[PPT] - On HI Banach spaces without reflexive subspaces and their spaces of PowerPoint Presentation

SLIDE 1

On HI Banach spaces without reflexive subspaces and their spaces of operator

Pavlos Motakis (joint work with S. Argyros)

Department of Mathematics Texas A&M University

Transfinite methods in Banach spaces and algebras of operators July 20, 2016 - B˛ edlewo

SLIDE 2

General discussion - L∞-spaces

Definition (J. Lindenstrauss - A. Pełczy´ nski (1968))

If X is separable a separable Banach space then X is called a L∞-space if there exists C 1 with X = ∪nEn, where (En)n is an increasing sequence of finite dimensional spaces with each En C-isomorphic to ℓdim En

∞

.

Typical examples of L∞-spaces: C(K) spaces,

e.g. C[0, 1], ℓ∞(N), or c0(N).

Non-typical examples of L∞-spaces:

constructed via the Bourgain-Delbaen construction method (1980), which was invented to present examples of L∞-spaces not containing c0.

SLIDE 4

General discussion - L∞-spaces

Definition (J. Lindenstrauss - A. Pełczy´ nski (1968))

If X is separable a separable Banach space then X is called a L∞-space if there exists C 1 with X = ∪nEn, where (En)n is an increasing sequence of finite dimensional spaces with each En C-isomorphic to ℓdim En

∞

.

Typical examples of L∞-spaces: C(K) spaces,

e.g. C[0, 1], ℓ∞(N), or c0(N).

Non-typical examples of L∞-spaces:

constructed via the Bourgain-Delbaen construction method (1980), which was invented to present examples of L∞-spaces not containing c0.

SLIDE 5

General discussion - L∞-spaces

Definition (J. Lindenstrauss - A. Pełczy´ nski (1968))

If X is separable a separable Banach space then X is called a L∞-space if there exists C 1 with X = ∪nEn, where (En)n is an increasing sequence of finite dimensional spaces with each En C-isomorphic to ℓdim En

∞

.

Typical examples of L∞-spaces: C(K) spaces,

e.g. C[0, 1], ℓ∞(N), or c0(N).

Non-typical examples of L∞-spaces:

constructed via the Bourgain-Delbaen construction method (1980), which was invented to present examples of L∞-spaces not containing c0.

SLIDE 6

General discussion - a problem under consideration

Problem (J. Bourgain (1981)

Does there exist a Banach space not containing c0, ℓ1, or reflexive subspaces?

Bourgain noted that the answer is unknown, even within the class
f L∞-spaces,

implying that such spaces that are moreover L∞ might not exist, i.e. that every L∞-space contains either c0, or ℓ1, or reflexive subspaces.

Why would he expect that?

SLIDE 7

General discussion - a problem under consideration

Problem (J. Bourgain (1981)

Does there exist a Banach space not containing c0, ℓ1, or reflexive subspaces?

Bourgain noted that the answer is unknown, even within the class
f L∞-spaces,

implying that such spaces that are moreover L∞ might not exist, i.e. that every L∞-space contains either c0, or ℓ1, or reflexive subspaces.

Why would he expect that?

SLIDE 8

General discussion - a problem under consideration

Problem (J. Bourgain (1981)

Does there exist a Banach space not containing c0, ℓ1, or reflexive subspaces?

Bourgain noted that the answer is unknown, even within the class
f L∞-spaces,

implying that such spaces that are moreover L∞ might not exist, i.e. that every L∞-space contains either c0, or ℓ1, or reflexive subspaces.

Why would he expect that?

SLIDE 9

General discussion - a problem under consideration

Problem (J. Bourgain (1981)

Does there exist a Banach space not containing c0, ℓ1, or reflexive subspaces?

Bourgain noted that the answer is unknown, even within the class
f L∞-spaces,

implying that such spaces that are moreover L∞ might not exist, i.e. that every L∞-space contains either c0, or ℓ1, or reflexive subspaces.

Why would he expect that?

SLIDE 10

Let’s investigate what a possible counter-example would look like.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

Any possible separable L∞-space X not containing ℓ1 or

reflexive subspaces cannot contain boundedly complete

sequences. explain why
If such an X also does not contain c0, then it contains no

unconditional sequences.

By Gowers’ dichotomy (1996) X must be HI saturated.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

SLIDE 11

Let’s investigate what a possible counter-example would look like.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

Any possible separable L∞-space X not containing ℓ1 or

reflexive subspaces cannot contain boundedly complete

sequences. explain why
If such an X also does not contain c0, then it contains no

unconditional sequences.

By Gowers’ dichotomy (1996) X must be HI saturated.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

SLIDE 12

Let’s investigate what a possible counter-example would look like.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

Any possible separable L∞-space X not containing ℓ1 or

reflexive subspaces cannot contain boundedly complete

sequences. explain why
If such an X also does not contain c0, then it contains no

unconditional sequences.

By Gowers’ dichotomy (1996) X must be HI saturated.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

SLIDE 13

Let’s investigate what a possible counter-example would look like.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

Any possible separable L∞-space X not containing ℓ1 or

reflexive subspaces cannot contain boundedly complete

sequences. explain why
If such an X also does not contain c0, then it contains no

unconditional sequences.

By Gowers’ dichotomy (1996) X must be HI saturated.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

SLIDE 14

Let’s investigate what a possible counter-example would look like.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

Any possible separable L∞-space X not containing ℓ1 or

reflexive subspaces cannot contain boundedly complete

sequences. explain why
If such an X also does not contain c0, then it contains no

unconditional sequences.

By Gowers’ dichotomy (1996) X must be HI saturated.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

SLIDE 15

Let’s investigate what a possible counter-example would look like.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

Any possible separable L∞-space X not containing ℓ1 or

reflexive subspaces cannot contain boundedly complete

sequences. explain why
If such an X also does not contain c0, then it contains no

unconditional sequences.

By Gowers’ dichotomy (1996) X must be HI saturated.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

SLIDE 16

Let’s investigate what a possible counter-example would look like.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

Any possible separable L∞-space X not containing ℓ1 or

reflexive subspaces cannot contain boundedly complete

sequences. explain why
If such an X also does not contain c0, then it contains no

unconditional sequences.

By Gowers’ dichotomy (1996) X must be HI saturated.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

SLIDE 17

General discussion - spaces with no reflexive subspaces

Recall Bourgain’s general problem: does there exist a space X not containing c0, ℓ1, or reflexive subspaces?

Theorem (W. T. Gowers (1994))

There exists a Banach space XG not containing c0, ℓ1, or reflexive subspaces.

The space XG is hereditarily indecomposable and all its infinite

subspaces have non-separable dual.

SLIDE 18

General discussion - spaces with no reflexive subspaces

Recall Bourgain’s general problem: does there exist a space X not containing c0, ℓ1, or reflexive subspaces?

Theorem (W. T. Gowers (1994))

There exists a Banach space XG not containing c0, ℓ1, or reflexive subspaces.

The space XG is hereditarily indecomposable and all its infinite

subspaces have non-separable dual.

SLIDE 19

General discussion - spaces with no reflexive subspaces

Recall Bourgain’s general problem: does there exist a space X not containing c0, ℓ1, or reflexive subspaces?

Theorem (W. T. Gowers (1994))

There exists a Banach space XG not containing c0, ℓ1, or reflexive subspaces.

The space XG is hereditarily indecomposable and all its infinite

subspaces have non-separable dual.

SLIDE 20

General discussion - L∞-spaces with no reflexive subspaces

Back to L∞-spaces: does there exist a L∞-space not containing

c0, ℓ1, or reflexive subspaces?

Possible approach: combine Gowers’ method of obtaining HI

spaces without reflexive subspaces and the Bourgain-Delbaen construction method.

Possible result: a separable L∞-space X with non-separable dual.
By Stegall: X contains ℓ1. This is undesired.
Was perhaps Bourgain rights and such spaces don’t exist?

SLIDE 21

General discussion - L∞-spaces with no reflexive subspaces

Back to L∞-spaces: does there exist a L∞-space not containing

c0, ℓ1, or reflexive subspaces?

Possible approach: combine Gowers’ method of obtaining HI

spaces without reflexive subspaces and the Bourgain-Delbaen construction method.

Possible result: a separable L∞-space X with non-separable dual.
By Stegall: X contains ℓ1. This is undesired.
Was perhaps Bourgain rights and such spaces don’t exist?

SLIDE 22

General discussion - L∞-spaces with no reflexive subspaces

Back to L∞-spaces: does there exist a L∞-space not containing

c0, ℓ1, or reflexive subspaces?

Possible approach: combine Gowers’ method of obtaining HI

spaces without reflexive subspaces and the Bourgain-Delbaen construction method.

Possible result: a separable L∞-space X with non-separable dual.
By Stegall: X contains ℓ1. This is undesired.
Was perhaps Bourgain rights and such spaces don’t exist?

SLIDE 23

General discussion - L∞-spaces with no reflexive subspaces

Back to L∞-spaces: does there exist a L∞-space not containing

c0, ℓ1, or reflexive subspaces?

Possible approach: combine Gowers’ method of obtaining HI

spaces without reflexive subspaces and the Bourgain-Delbaen construction method.

Possible result: a separable L∞-space X with non-separable dual.
By Stegall: X contains ℓ1. This is undesired.
Was perhaps Bourgain rights and such spaces don’t exist?

SLIDE 24

General discussion - L∞-spaces with no reflexive subspaces

Back to L∞-spaces: does there exist a L∞-space not containing

c0, ℓ1, or reflexive subspaces?

Possible approach: combine Gowers’ method of obtaining HI

spaces without reflexive subspaces and the Bourgain-Delbaen construction method.

Possible result: a separable L∞-space X with non-separable dual.
By Stegall: X contains ℓ1. This is undesired.
Was perhaps Bourgain rights and such spaces don’t exist?

SLIDE 25

An answer to this problem

Theorem (S. Argyros - M)

There exists a separable L∞-space Xnr that is HI (and hence does not contain c0 or ℓ1), has separable dual and does not have reflexive

subspaces. Moreover, this space has the scalar-plus-compact

property.

Definition

A Banach space X has the scalar-plus compact if every bounded linear operator T : X → X is of the form T = λI + K with λ ∈ R and K a compact operator.

SLIDE 26

An answer to this problem

Theorem (S. Argyros - M)

There exists a separable L∞-space Xnr that is HI (and hence does not contain c0 or ℓ1), has separable dual and does not have reflexive

subspaces. Moreover, this space has the scalar-plus-compact

property.

Definition

A Banach space X has the scalar-plus compact if every bounded linear operator T : X → X is of the form T = λI + K with λ ∈ R and K a compact operator.

SLIDE 27

Some remarks

Bourgain’s problem in the class of L∞-spaces can be related to

the following.

Problem (H. P . Rosenthal)

Let X be a L∞-saturated space. Does X contain c0?

Reversing the problem: does there exists a L∞-saturated space X

not containing c0?

If such a (separable L∞-space) X exists, then it would not contain

ℓ1, c0, or reflexive subspaces

X would be a counter-example to Bourgain’s problem in the class
f L∞-spaces.
Is Xnr such an X?

It is not.

SLIDE 28

Some remarks

Bourgain’s problem in the class of L∞-spaces can be related to

the following.

Problem (H. P . Rosenthal)

Let X be a L∞-saturated space. Does X contain c0?

Reversing the problem: does there exists a L∞-saturated space X

not containing c0?

If such a (separable L∞-space) X exists, then it would not contain

ℓ1, c0, or reflexive subspaces

X would be a counter-example to Bourgain’s problem in the class
f L∞-spaces.
Is Xnr such an X?

It is not.

SLIDE 29

Some remarks

Bourgain’s problem in the class of L∞-spaces can be related to

the following.

Problem (H. P . Rosenthal)

Let X be a L∞-saturated space. Does X contain c0?

Reversing the problem: does there exists a L∞-saturated space X

not containing c0?

If such a (separable L∞-space) X exists, then it would not contain

ℓ1, c0, or reflexive subspaces

X would be a counter-example to Bourgain’s problem in the class
f L∞-spaces.
Is Xnr such an X?

It is not.

SLIDE 30

Some remarks

Bourgain’s problem in the class of L∞-spaces can be related to

the following.

Problem (H. P . Rosenthal)

Let X be a L∞-saturated space. Does X contain c0?

Reversing the problem: does there exists a L∞-saturated space X

not containing c0?

If such a (separable L∞-space) X exists, then it would not contain

ℓ1, c0, or reflexive subspaces

X would be a counter-example to Bourgain’s problem in the class
f L∞-spaces.
Is Xnr such an X?

It is not.

SLIDE 31

Some remarks

Bourgain’s problem in the class of L∞-spaces can be related to

the following.

Problem (H. P . Rosenthal)

Let X be a L∞-saturated space. Does X contain c0?

Reversing the problem: does there exists a L∞-saturated space X

not containing c0?

If such a (separable L∞-space) X exists, then it would not contain

ℓ1, c0, or reflexive subspaces

X would be a counter-example to Bourgain’s problem in the class
f L∞-spaces.
Is Xnr such an X?

It is not.

SLIDE 32

Some remarks

Bourgain’s problem in the class of L∞-spaces can be related to

the following.

Problem (H. P . Rosenthal)

Let X be a L∞-saturated space. Does X contain c0?

Reversing the problem: does there exists a L∞-saturated space X

not containing c0?

If such a (separable L∞-space) X exists, then it would not contain

ℓ1, c0, or reflexive subspaces

X would be a counter-example to Bourgain’s problem in the class
f L∞-spaces.
Is Xnr such an X?

It is not.

SLIDE 33

Some remarks

Bourgain’s problem in the class of L∞-spaces can be related to

the following.

Problem (H. P . Rosenthal)

Let X be a L∞-saturated space. Does X contain c0?

Reversing the problem: does there exists a L∞-saturated space X

not containing c0?

If such a (separable L∞-space) X exists, then it would not contain

ℓ1, c0, or reflexive subspaces

X would be a counter-example to Bourgain’s problem in the class
f L∞-spaces.
Is Xnr such an X?

It is not.

SLIDE 34

Some remarks on the construction of Xnr

The space Xnr is constructed with the Bourgain-Delbaen method,

but not in combination with Gowers’ method.

Instead, it uses a new variation of the method of saturation under

constraints, a Tsirelson-type saturation method.

This version of saturation under constraints provides a new way of

defining HI spaces, where the conditional structure is provided by certain averages, and not special functionals. Intereasting feature: it provides a unified approach for constructing reflexive HI spaces and HI spaces with no boundedly complete sequences.

SLIDE 35

Some remarks on the construction of Xnr

The space Xnr is constructed with the Bourgain-Delbaen method,

but not in combination with Gowers’ method.

Instead, it uses a new variation of the method of saturation under

constraints, a Tsirelson-type saturation method.

This version of saturation under constraints provides a new way of

defining HI spaces, where the conditional structure is provided by certain averages, and not special functionals. Intereasting feature: it provides a unified approach for constructing reflexive HI spaces and HI spaces with no boundedly complete sequences.

SLIDE 36

Some remarks on the construction of Xnr

The space Xnr is constructed with the Bourgain-Delbaen method,

but not in combination with Gowers’ method.

Instead, it uses a new variation of the method of saturation under

constraints, a Tsirelson-type saturation method.

This version of saturation under constraints provides a new way of

defining HI spaces, where the conditional structure is provided by certain averages, and not special functionals. Intereasting feature: it provides a unified approach for constructing reflexive HI spaces and HI spaces with no boundedly complete sequences.

SLIDE 37

Some remarks on the construction of Xnr

The space Xnr is constructed with the Bourgain-Delbaen method,

but not in combination with Gowers’ method.

Instead, it uses a new variation of the method of saturation under

constraints, a Tsirelson-type saturation method.

This version of saturation under constraints provides a new way of

defining HI spaces, where the conditional structure is provided by certain averages, and not special functionals. Intereasting feature: it provides a unified approach for constructing reflexive HI spaces and HI spaces with no boundedly complete sequences.

SLIDE 38

Some results of new saturation under constraints

A result based on this new method in a classical (Tsirelson,

non-L∞) setting.

Theorem (S. Argyros - M. (2015))

There exists a hereditarily indecomposable Banach space XU that has separable dual and does not contain boundedly complete

sequences. In particular is has no reflexive subspaces.

(i) Every bounded linear operator T defined on XU is of the form T = λI + S with S strictly singular. (ii) The ideal of strictly singular operators on XU is non-separable. (iii) The composition of any two strictly singular operators defined on XU is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space XT satisfying (i), (ii), (iii), and (iv).

SLIDE 39

Some results of new saturation under constraints

A result based on this new method in a classical (Tsirelson,

non-L∞) setting.

Theorem (S. Argyros - M. (2015))

There exists a hereditarily indecomposable Banach space XU that has separable dual and does not contain boundedly complete

sequences. In particular is has no reflexive subspaces.

(i) Every bounded linear operator T defined on XU is of the form T = λI + S with S strictly singular. (ii) The ideal of strictly singular operators on XU is non-separable. (iii) The composition of any two strictly singular operators defined on XU is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space XT satisfying (i), (ii), (iii), and (iv).

SLIDE 40

Some results of new saturation under constraints

A result based on this new method in a classical (Tsirelson,

non-L∞) setting.

Theorem (S. Argyros - M. (2015))

There exists a hereditarily indecomposable Banach space XU that has separable dual and does not contain boundedly complete

sequences. In particular is has no reflexive subspaces.

(i) Every bounded linear operator T defined on XU is of the form T = λI + S with S strictly singular. (ii) The ideal of strictly singular operators on XU is non-separable. (iii) The composition of any two strictly singular operators defined on XU is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space XT satisfying (i), (ii), (iii), and (iv).

SLIDE 41

Some results of new saturation under constraints

A result based on this new method in a classical (Tsirelson,

non-L∞) setting.

Theorem (S. Argyros - M. (2015))

There exists a hereditarily indecomposable Banach space XU that has separable dual and does not contain boundedly complete

sequences. In particular is has no reflexive subspaces.

(i) Every bounded linear operator T defined on XU is of the form T = λI + S with S strictly singular. (ii) The ideal of strictly singular operators on XU is non-separable. (iii) The composition of any two strictly singular operators defined on XU is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space XT satisfying (i), (ii), (iii), and (iv).

SLIDE 42

Some results of new saturation under constraints

A result based on this new method in a classical (Tsirelson,

non-L∞) setting.

Theorem (S. Argyros - M. (2015))

There exists a hereditarily indecomposable Banach space XU that has separable dual and does not contain boundedly complete

sequences. In particular is has no reflexive subspaces.

(i) Every bounded linear operator T defined on XU is of the form T = λI + S with S strictly singular. (ii) The ideal of strictly singular operators on XU is non-separable. (iii) The composition of any two strictly singular operators defined on XU is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space XT satisfying (i), (ii), (iii), and (iv).

SLIDE 43

Some results of new saturation under constraints

A result based on this new method in a classical (Tsirelson,

non-L∞) setting.

Theorem (S. Argyros - M. (2015))

There exists a hereditarily indecomposable Banach space XU that has separable dual and does not contain boundedly complete

sequences. In particular is has no reflexive subspaces.

(i) Every bounded linear operator T defined on XU is of the form T = λI + S with S strictly singular. (ii) The ideal of strictly singular operators on XU is non-separable. (iii) The composition of any two strictly singular operators defined on XU is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space XT satisfying (i), (ii), (iii), and (iv).

SLIDE 44

Some results of new saturation under constraints

A result based on this new method in a classical (Tsirelson,

non-L∞) setting.

Theorem (S. Argyros - M. (2015))

There exists a hereditarily indecomposable Banach space XU that has separable dual and does not contain boundedly complete

sequences. In particular is has no reflexive subspaces.

(i) Every bounded linear operator T defined on XU is of the form T = λI + S with S strictly singular. (ii) The ideal of strictly singular operators on XU is non-separable. (iii) The composition of any two strictly singular operators defined on XU is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space XT satisfying (i), (ii), (iii), and (iv).

SLIDE 45

Some results of new saturation under constraints

A nice application:

Definition

Let X be a Banach space. The Calkin algebra of X is the quotient algebra Cαℓ(X) = L(X)/K(X).

Theorem (R. Skillicorn (2015))

The Calkin algebra of XU is non-separable and it admits two non-equivalent complete algebra norms.

The above uses that Cαℓ(XU) = (R[Id]) ⊕ (S(XU)/K(XU)) and

that for every [S], [T] ∈ S(XU)/K(XU) we have [S][T] = 0.

SLIDE 46

Some results of new saturation under constraints

A nice application:

Definition

Let X be a Banach space. The Calkin algebra of X is the quotient algebra Cαℓ(X) = L(X)/K(X).

Theorem (R. Skillicorn (2015))

The Calkin algebra of XU is non-separable and it admits two non-equivalent complete algebra norms.

The above uses that Cαℓ(XU) = (R[Id]) ⊕ (S(XU)/K(XU)) and

that for every [S], [T] ∈ S(XU)/K(XU) we have [S][T] = 0.

SLIDE 47

Some results of new saturation under constraints

A nice application:

Definition

Let X be a Banach space. The Calkin algebra of X is the quotient algebra Cαℓ(X) = L(X)/K(X).

Theorem (R. Skillicorn (2015))

The Calkin algebra of XU is non-separable and it admits two non-equivalent complete algebra norms.

The above uses that Cαℓ(XU) = (R[Id]) ⊕ (S(XU)/K(XU)) and

that for every [S], [T] ∈ S(XU)/K(XU) we have [S][T] = 0.

SLIDE 48

Saturation under constraints - some history Method of saturation under constraints

Most important feature: allows heterogeneous asymptotic

structure to appear hereditarily.

It was used for the first time by E. Odell and Th. Schlumprecht

and it was further developed by S. Argyros and his collaborators (e.g. Beanland, Freeman, M. etc)

It can be used to address problems concerning local block

structures, asymptotic structures, and operators defined on certain spaces.

SLIDE 49

Saturation under constraints - some history Method of saturation under constraints

Most important feature: allows heterogeneous asymptotic

structure to appear hereditarily.

It was used for the first time by E. Odell and Th. Schlumprecht

and it was further developed by S. Argyros and his collaborators (e.g. Beanland, Freeman, M. etc)

It can be used to address problems concerning local block

structures, asymptotic structures, and operators defined on certain spaces.

SLIDE 50

Saturation under constraints - some history Method of saturation under constraints

Most important feature: allows heterogeneous asymptotic

structure to appear hereditarily.

It was used for the first time by E. Odell and Th. Schlumprecht

and it was further developed by S. Argyros and his collaborators (e.g. Beanland, Freeman, M. etc)

It can be used to address problems concerning local block

structures, asymptotic structures, and operators defined on certain spaces.

SLIDE 51

Saturation under constraints - some history Method of saturation under constraints

Most important feature: allows heterogeneous asymptotic

structure to appear hereditarily.

It was used for the first time by E. Odell and Th. Schlumprecht

and it was further developed by S. Argyros and his collaborators (e.g. Beanland, Freeman, M. etc)

It can be used to address problems concerning local block

structures, asymptotic structures, and operators defined on certain spaces.

SLIDE 52

Saturation under constraints - norming sets

A way of defining norms: norming sets
If X is a vector space and W ⊂ X #, W is called a norming set if the

function · W : X → R with xW = sup{|f(x)| : f ∈ W} is a norm.

Commonly, X = c00(N) and W ⊂ c00(N).
This makes sense: if x = (ak)k, f = (bk)k are in c00(N) then

f(x) =

∞

k=1

akbk (this is a finite sum).

SLIDE 53

Saturation under constraints - norming sets

A way of defining norms: norming sets
If X is a vector space and W ⊂ X #, W is called a norming set if the

function · W : X → R with xW = sup{|f(x)| : f ∈ W} is a norm.

Commonly, X = c00(N) and W ⊂ c00(N).
This makes sense: if x = (ak)k, f = (bk)k are in c00(N) then

f(x) =

∞

k=1

akbk (this is a finite sum).

SLIDE 54

Saturation under constraints - norming sets

A way of defining norms: norming sets
If X is a vector space and W ⊂ X #, W is called a norming set if the

function · W : X → R with xW = sup{|f(x)| : f ∈ W} is a norm.

Commonly, X = c00(N) and W ⊂ c00(N).
This makes sense: if x = (ak)k, f = (bk)k are in c00(N) then

f(x) =

∞

k=1

akbk (this is a finite sum).

SLIDE 55

Saturation under constraints - norming sets

A way of defining norms: norming sets
If X is a vector space and W ⊂ X #, W is called a norming set if the

function · W : X → R with xW = sup{|f(x)| : f ∈ W} is a norm.

Commonly, X = c00(N) and W ⊂ c00(N).
This makes sense: if x = (ak)k, f = (bk)k are in c00(N) then

f(x) =

∞

k=1

akbk (this is a finite sum).

SLIDE 56

Saturation under constraints - norming sets

A way of defining norms: norming sets
If X is a vector space and W ⊂ X #, W is called a norming set if the

function · W : X → R with xW = sup{|f(x)| : f ∈ W} is a norm.

Commonly, X = c00(N) and W ⊂ c00(N).
This makes sense: if x = (ak)k, f = (bk)k are in c00(N) then

f(x) =

∞

k=1

akbk (this is a finite sum).

SLIDE 57

Saturation under constraints - norming sets

Two vectors x, y are called successive (x < y) if

max supp(x) < min supp(y), where their supports are taken with respect to the unit vector basis of c00(N)

The Schreier familiers:

S1 = {F ⊂ N : #F min(F)} Sn+1 = {∪m

k=1Fk : m ∈ N and m F1 < · · · < Fm} .

A sequence of vectors x1, x2, · · · , xm is Sn-admissible if

x1 < x2 < · · · < xm and {min supp(xk) : 1 k m} ∈ Sn.

In particular, x1 < · · · < xn are S1-admissible means

n min supp(x1).

SLIDE 58

Saturation under constraints - norming sets

Two vectors x, y are called successive (x < y) if

max supp(x) < min supp(y), where their supports are taken with respect to the unit vector basis of c00(N)

The Schreier familiers:

S1 = {F ⊂ N : #F min(F)} Sn+1 = {∪m

k=1Fk : m ∈ N and m F1 < · · · < Fm} .

A sequence of vectors x1, x2, · · · , xm is Sn-admissible if

x1 < x2 < · · · < xm and {min supp(xk) : 1 k m} ∈ Sn.

In particular, x1 < · · · < xn are S1-admissible means

n min supp(x1).

SLIDE 59

Saturation under constraints - norming sets

Two vectors x, y are called successive (x < y) if

max supp(x) < min supp(y), where their supports are taken with respect to the unit vector basis of c00(N)

The Schreier familiers:

S1 = {F ⊂ N : #F min(F)} Sn+1 = {∪m

k=1Fk : m ∈ N and m F1 < · · · < Fm} .

A sequence of vectors x1, x2, · · · , xm is Sn-admissible if

x1 < x2 < · · · < xm and {min supp(xk) : 1 k m} ∈ Sn.

In particular, x1 < · · · < xn are S1-admissible means

n min supp(x1).

SLIDE 60

Saturation under constraints - norming sets

Two vectors x, y are called successive (x < y) if

max supp(x) < min supp(y), where their supports are taken with respect to the unit vector basis of c00(N)

The Schreier familiers:

S1 = {F ⊂ N : #F min(F)} Sn+1 = {∪m

k=1Fk : m ∈ N and m F1 < · · · < Fm} .

A sequence of vectors x1, x2, · · · , xm is Sn-admissible if

x1 < x2 < · · · < xm and {min supp(xk) : 1 k m} ∈ Sn.

In particular, x1 < · · · < xn are S1-admissible means

n min supp(x1).

SLIDE 61

Saturation under constraints - norming sets

Two vectors x, y are called successive (x < y) if

max supp(x) < min supp(y), where their supports are taken with respect to the unit vector basis of c00(N)

The Schreier familiers:

S1 = {F ⊂ N : #F min(F)} Sn+1 = {∪m

k=1Fk : m ∈ N and m F1 < · · · < Fm} .

A sequence of vectors x1, x2, · · · , xm is Sn-admissible if

x1 < x2 < · · · < xm and {min supp(xk) : 1 k m} ∈ Sn.

In particular, x1 < · · · < xn are S1-admissible means

n min supp(x1).

SLIDE 62

Saturation under constraints - norming sets

Two vectors x, y are called successive (x < y) if

max supp(x) < min supp(y), where their supports are taken with respect to the unit vector basis of c00(N)

The Schreier familiers:

S1 = {F ⊂ N : #F min(F)} Sn+1 = {∪m

k=1Fk : m ∈ N and m F1 < · · · < Fm} .

A sequence of vectors x1, x2, · · · , xm is Sn-admissible if

x1 < x2 < · · · < xm and {min supp(xk) : 1 k m} ∈ Sn.

In particular, x1 < · · · < xn are S1-admissible means

n min supp(x1).

SLIDE 63

Saturated norms - Tsirelson space

First, a typical norming set.
We define WT = ∪∞

n=0Wn where W0 ⊂ W1 ⊂ W2 ⊂ · · · .

Set W0 = {±ei : i ∈ N} and Wn+1 = Wn ∪

1

2

m

k=1

fk : d ∈ N and (fk)m

k=1 is an

draw

picture

S1-admissible sequence of Wn
.
If T is the completion of c00(N) endowed with · WT , then T is

Tsirelson space.

SLIDE 64

Saturated norms - Tsirelson space

First, a typical norming set.
We define WT = ∪∞

n=0Wn where W0 ⊂ W1 ⊂ W2 ⊂ · · · .

Set W0 = {±ei : i ∈ N} and Wn+1 = Wn ∪

1

2

m

k=1

fk : d ∈ N and (fk)m

k=1 is an

draw

picture

S1-admissible sequence of Wn
.
If T is the completion of c00(N) endowed with · WT , then T is

Tsirelson space.

SLIDE 65

Saturated norms - Tsirelson space

First, a typical norming set.
We define WT = ∪∞

n=0Wn where W0 ⊂ W1 ⊂ W2 ⊂ · · · .

Set W0 = {±ei : i ∈ N} and Wn+1 = Wn ∪

1

2

m

k=1

fk : d ∈ N and (fk)m

k=1 is an

draw

picture

S1-admissible sequence of Wn
.
If T is the completion of c00(N) endowed with · WT , then T is

Tsirelson space.

SLIDE 66

Saturated norms - Tsirelson space

First, a typical norming set.
We define WT = ∪∞

n=0Wn where W0 ⊂ W1 ⊂ W2 ⊂ · · · .

Set W0 = {±ei : i ∈ N} and Wn+1 = Wn ∪

1

2

m

k=1

fk : d ∈ N and (fk)m

k=1 is an

draw

picture

S1-admissible sequence of Wn
.
If T is the completion of c00(N) endowed with · WT , then T is

Tsirelson space.

SLIDE 67

Saturated norms - Tsirelson space

Such constructions always yield boundedly complete bases, e.g.

Tsirelson space has the stronger property of being asymptotic-ℓ1.

That is, for S1-admissible vector x1 < · · · < xm:
m
k=1

xk

T

1 2

m

k=1

xkT.

Because, for any S1-admissible f1 < · · · < fm in W we have

(1/2) m

k=1 fk is in W. (picture)

SLIDE 68

Saturated norms - Tsirelson space

Such constructions always yield boundedly complete bases, e.g.

Tsirelson space has the stronger property of being asymptotic-ℓ1.

That is, for S1-admissible vector x1 < · · · < xm:
m
k=1

xk

T

1 2

m

k=1

xkT.

Because, for any S1-admissible f1 < · · · < fm in W we have

(1/2) m

k=1 fk is in W. (picture)

SLIDE 69

Saturated norms - Tsirelson space

Such constructions always yield boundedly complete bases, e.g.

Tsirelson space has the stronger property of being asymptotic-ℓ1.

That is, for S1-admissible vector x1 < · · · < xm:
m
k=1

xk

T

1 2

m

k=1

xkT.

Because, for any S1-admissible f1 < · · · < fm in W we have

(1/2) m

k=1 fk is in W. (picture)

SLIDE 70

Saturation under constraints - the norming set Wα

We now define a simple norming set saturated under constraints.
Let G be a subset of c00(N). A vector α0 ∈ c00(N) is called an

α-average of G of size s(α0) = n if there are f1 < · · · < fn in G with α0 = 1 n (f1 + · · · + fn) .

A sequence of α-averages α1 < α2 < · · · is called very fast

growing if s(αk+1) > max supp(αk).

SLIDE 71

Saturation under constraints - the norming set Wα

We now define a simple norming set saturated under constraints.
Let G be a subset of c00(N). A vector α0 ∈ c00(N) is called an

α-average of G of size s(α0) = n if there are f1 < · · · < fn in G with α0 = 1 n (f1 + · · · + fn) .

A sequence of α-averages α1 < α2 < · · · is called very fast

growing if s(αk+1) > max supp(αk).

SLIDE 72

Saturation under constraints - the norming set Wα

We now define a simple norming set saturated under constraints.
Let G be a subset of c00(N). A vector α0 ∈ c00(N) is called an

α-average of G of size s(α0) = n if there are f1 < · · · < fn in G with α0 = 1 n (f1 + · · · + fn) .

A sequence of α-averages α1 < α2 < · · · is called very fast

growing if s(αk+1) > max supp(αk).

SLIDE 73

Saturation under constraints - the norming set Wα

We define Wα = ∪∞

n=0W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · .

Set W α

0 = {±ei : i ∈ N} and

W α

n+1 = W α n ∪

1

2d

m

k=1

αk : d ∈ N and (αk)m

k=1 is an Sd-admissible

sequence of very fast growing α-averages of Wn

.
If T(1/2n,Sn,α) is the completion of c00(N) endowed with · W, then

T(1/2n,Sn,α) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and Sd-admissible and very fast growing α-averages α1 < · · · < αm in Wα, f = (1/2d) m

k=1 αk is in Wα.

Such an f is said to have weight d.

SLIDE 74

Saturation under constraints - the norming set Wα

We define Wα = ∪∞

n=0W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · .

Set W α

0 = {±ei : i ∈ N} and

W α

n+1 = W α n ∪

1

2d

m

k=1

αk : d ∈ N and (αk)m

k=1 is an Sd-admissible

sequence of very fast growing α-averages of Wn

.
If T(1/2n,Sn,α) is the completion of c00(N) endowed with · W, then

T(1/2n,Sn,α) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and Sd-admissible and very fast growing α-averages α1 < · · · < αm in Wα, f = (1/2d) m

k=1 αk is in Wα.

Such an f is said to have weight d.

SLIDE 75

Saturation under constraints - the norming set Wα

We define Wα = ∪∞

n=0W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · .

Set W α

0 = {±ei : i ∈ N} and

W α

n+1 = W α n ∪

1

2d

m

k=1

αk : d ∈ N and (αk)m

k=1 is an Sd-admissible

sequence of very fast growing α-averages of Wn

.
If T(1/2n,Sn,α) is the completion of c00(N) endowed with · W, then

T(1/2n,Sn,α) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and Sd-admissible and very fast growing α-averages α1 < · · · < αm in Wα, f = (1/2d) m

k=1 αk is in Wα.

Such an f is said to have weight d.

SLIDE 76

Saturation under constraints - the norming set Wα

We define Wα = ∪∞

n=0W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · .

Set W α

0 = {±ei : i ∈ N} and

W α

n+1 = W α n ∪

1

2d

m

k=1

αk : d ∈ N and (αk)m

k=1 is an Sd-admissible

sequence of very fast growing α-averages of Wn

.
If T(1/2n,Sn,α) is the completion of c00(N) endowed with · W, then

T(1/2n,Sn,α) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and Sd-admissible and very fast growing α-averages α1 < · · · < αm in Wα, f = (1/2d) m

k=1 αk is in Wα.

Such an f is said to have weight d.

SLIDE 77

Saturation under constraints - the norming set Wα

We define Wα = ∪∞

n=0W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · .

Set W α

0 = {±ei : i ∈ N} and

W α

n+1 = W α n ∪

1

2d

m

k=1

αk : d ∈ N and (αk)m

k=1 is an Sd-admissible

sequence of very fast growing α-averages of Wn

.
If T(1/2n,Sn,α) is the completion of c00(N) endowed with · W, then

T(1/2n,Sn,α) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and Sd-admissible and very fast growing α-averages α1 < · · · < αm in Wα, f = (1/2d) m

k=1 αk is in Wα.

Such an f is said to have weight d.

SLIDE 78

Saturation under constraints - the space T(1/2n,Sn,α)

The space T(1/2n,Sn,α) is reflexive and it has an unconditional

basis(that is boundedly complete).

Every normalized block sequence in Tα has a subsequence that

admits an ℓ1 spreading model or a subsequence that admits a c0-spreading model. Both of these types of sequences exist in every block subspace of T(1/2n,Sn,α).

Definition

Let (xk)k be a sequence in some Banach space. We say that (xk)k generates a c0 spreading model if there exists C 1 so that for every n k1 < · · · < kn the sequence (xki)n

i=1 is C-equivalent to the unit

vector basis of ℓn

∞.

Other spreading models, e.g. ℓ1 or the summing basis of c0 are similarly defined.

SLIDE 79

Saturation under constraints - the space T(1/2n,Sn,α)

The space T(1/2n,Sn,α) is reflexive and it has an unconditional

basis(that is boundedly complete).

Every normalized block sequence in Tα has a subsequence that

admits an ℓ1 spreading model or a subsequence that admits a c0-spreading model. Both of these types of sequences exist in every block subspace of T(1/2n,Sn,α).

Definition

Let (xk)k be a sequence in some Banach space. We say that (xk)k generates a c0 spreading model if there exists C 1 so that for every n k1 < · · · < kn the sequence (xki)n

i=1 is C-equivalent to the unit

vector basis of ℓn

∞.

Other spreading models, e.g. ℓ1 or the summing basis of c0 are similarly defined.

SLIDE 80

Saturation under constraints - the space T(1/2n,Sn,α)

The space T(1/2n,Sn,α) is reflexive and it has an unconditional

basis(that is boundedly complete).

Every normalized block sequence in Tα has a subsequence that

admits an ℓ1 spreading model or a subsequence that admits a c0-spreading model. Both of these types of sequences exist in every block subspace of T(1/2n,Sn,α).

Definition

Let (xk)k be a sequence in some Banach space. We say that (xk)k generates a c0 spreading model if there exists C 1 so that for every n k1 < · · · < kn the sequence (xki)n

i=1 is C-equivalent to the unit

vector basis of ℓn

∞.

Other spreading models, e.g. ℓ1 or the summing basis of c0 are similarly defined.

SLIDE 81

Saturation under constraints - new method

To avoid having a boundedly complete basis, the new variation of

saturation under constraints restricts the choice of averages allowed to be used in the norming set.

The choice is made with the help of an appropriate tree.
Denote Q the set of all finite sequences (fk, xk)n

k=1 with

f1 < · · · < fn ∈ Wα and xk vectors in c00(N) with rational coefficients.

Using a coding function, choose a subtree U of Q so that for all

(fk, xk)n

k=1 in U, the weight of fn uniquely determines the sequence

(fk, xk)n−1

k=1.

This tree is ill-founded, every maximal chain is infinite. It is used to

define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.

SLIDE 82

Saturation under constraints - new method

To avoid having a boundedly complete basis, the new variation of

saturation under constraints restricts the choice of averages allowed to be used in the norming set.

The choice is made with the help of an appropriate tree.
Denote Q the set of all finite sequences (fk, xk)n

k=1 with

f1 < · · · < fn ∈ Wα and xk vectors in c00(N) with rational coefficients.

Using a coding function, choose a subtree U of Q so that for all

(fk, xk)n

k=1 in U, the weight of fn uniquely determines the sequence

(fk, xk)n−1

k=1.

This tree is ill-founded, every maximal chain is infinite. It is used to

define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.

SLIDE 83

Saturation under constraints - new method

To avoid having a boundedly complete basis, the new variation of

saturation under constraints restricts the choice of averages allowed to be used in the norming set.

The choice is made with the help of an appropriate tree.
Denote Q the set of all finite sequences (fk, xk)n

k=1 with

f1 < · · · < fn ∈ Wα and xk vectors in c00(N) with rational coefficients.

Using a coding function, choose a subtree U of Q so that for all

(fk, xk)n

k=1 in U, the weight of fn uniquely determines the sequence

(fk, xk)n−1

k=1.

This tree is ill-founded, every maximal chain is infinite. It is used to

define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.

SLIDE 84

Saturation under constraints - new method

To avoid having a boundedly complete basis, the new variation of

saturation under constraints restricts the choice of averages allowed to be used in the norming set.

The choice is made with the help of an appropriate tree.
Denote Q the set of all finite sequences (fk, xk)n

k=1 with

f1 < · · · < fn ∈ Wα and xk vectors in c00(N) with rational coefficients.

Using a coding function, choose a subtree U of Q so that for all

(fk, xk)n

k=1 in U, the weight of fn uniquely determines the sequence

(fk, xk)n−1

k=1.

This tree is ill-founded, every maximal chain is infinite. It is used to

define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.

SLIDE 85

Saturation under constraints - new method

To avoid having a boundedly complete basis, the new variation of

saturation under constraints restricts the choice of averages allowed to be used in the norming set.

The choice is made with the help of an appropriate tree.
Denote Q the set of all finite sequences (fk, xk)n

k=1 with

f1 < · · · < fn ∈ Wα and xk vectors in c00(N) with rational coefficients.

Using a coding function, choose a subtree U of Q so that for all

(fk, xk)n

k=1 in U, the weight of fn uniquely determines the sequence

(fk, xk)n−1

k=1.

This tree is ill-founded, every maximal chain is infinite. It is used to

define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.

SLIDE 86

Saturation under constraints - new method

To avoid having a boundedly complete basis, the new variation of

saturation under constraints restricts the choice of averages allowed to be used in the norming set.

The choice is made with the help of an appropriate tree.
Denote Q the set of all finite sequences (fk, xk)n

k=1 with

f1 < · · · < fn ∈ Wα and xk vectors in c00(N) with rational coefficients.

Using a coding function, choose a subtree U of Q so that for all

(fk, xk)n

k=1 in U, the weight of fn uniquely determines the sequence

(fk, xk)n−1

k=1.

This tree is ill-founded, every maximal chain is infinite. It is used to

define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.

SLIDE 87

Saturation under constraints - new method

Given G ⊂ Wα, the tree U defines four types of α-averages, built on

elements of G and collectively called αc-averages of G.

One example: if (fk, xk)n

k=1 is in U so that fk ∈ G and

fk(xk) = fm(xm) for 1 k m n, then α0 = 1 n

n

k=1

(−1)kfk is an αc-average of G. Specifically, averages similar to the above are called compatible averages of G. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements f1 < · · · < fn
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 88

Saturation under constraints - new method

Given G ⊂ Wα, the tree U defines four types of α-averages, built on

elements of G and collectively called αc-averages of G.

One example: if (fk, xk)n

k=1 is in U so that fk ∈ G and

fk(xk) = fm(xm) for 1 k m n, then α0 = 1 n

n

k=1

(−1)kfk is an αc-average of G. Specifically, averages similar to the above are called compatible averages of G. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements f1 < · · · < fn
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 89

Saturation under constraints - new method

Given G ⊂ Wα, the tree U defines four types of α-averages, built on

elements of G and collectively called αc-averages of G.

One example: if (fk, xk)n

k=1 is in U so that fk ∈ G and

fk(xk) = fm(xm) for 1 k m n, then α0 = 1 n

n

k=1

(−1)kfk is an αc-average of G. Specifically, averages similar to the above are called compatible averages of G. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements f1 < · · · < fn
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 90

Saturation under constraints - new method

Given G ⊂ Wα, the tree U defines four types of α-averages, built on

elements of G and collectively called αc-averages of G.

One example: if (fk, xk)n

k=1 is in U so that fk ∈ G and

fk(xk) = fm(xm) for 1 k m n, then α0 = 1 n

n

k=1

(−1)kfk is an αc-average of G. Specifically, averages similar to the above are called compatible averages of G. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements f1 < · · · < fn
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 91

Saturation under constraints - new method

Given G ⊂ Wα, the tree U defines four types of α-averages, built on

elements of G and collectively called αc-averages of G.

One example: if (fk, xk)n

k=1 is in U so that fk ∈ G and

fk(xk) = fm(xm) for 1 k m n, then α0 = 1 n

n

k=1

(−1)kfk is an αc-average of G. Specifically, averages similar to the above are called compatible averages of G. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements f1 < · · · < fn
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 92

Saturation under constraints - new method

We choose a norming set WU to be the smallest subset of Wα that
contains the unit vectors ei and is symmetric and
for any n ∈ N and Sn-admissible and very fast growing sequence

(αq)d

q=1 of αc-averages of WU the functional

f = 1 2n

d

q=1

αq is in WU.

We denote the completion of (c00(N), · WU ) as XU.

SLIDE 93

Saturation under constraints - new method

We choose a norming set WU to be the smallest subset of Wα that
contains the unit vectors ei and is symmetric and
for any n ∈ N and Sn-admissible and very fast growing sequence

(αq)d

q=1 of αc-averages of WU the functional

f = 1 2n

d

q=1

αq is in WU.

We denote the completion of (c00(N), · WU ) as XU.

SLIDE 94

Saturation under constraints - new method

We choose a norming set WU to be the smallest subset of Wα that
contains the unit vectors ei and is symmetric and
for any n ∈ N and Sn-admissible and very fast growing sequence

(αq)d

q=1 of αc-averages of WU the functional

f = 1 2n

d

q=1

αq is in WU.

We denote the completion of (c00(N), · WU ) as XU.

SLIDE 95

Saturation under constraints - new method

We choose a norming set WU to be the smallest subset of Wα that
contains the unit vectors ei and is symmetric and
for any n ∈ N and Sn-admissible and very fast growing sequence

(αq)d

q=1 of αc-averages of WU the functional

f = 1 2n

d

q=1

αq is in WU.

We denote the completion of (c00(N), · WU ) as XU.

SLIDE 96

Saturation under constraints - new method

Properties of the space XU

◮ It is hereditarily indecomposable, it has separable dual, and it

has no boundedly complete sequences.

◮ Every Schauder basic sequence in XU admits either c0, or ℓ1, or

the summing basis of c0 as a spreading model. All three type of spreading models appear in every subspace of XU.

◮ For every bounded linear operator T defined on XU there is a

scalar λ so that S = λI − T is weakly compact and hence strictly singular.

◮ The composition of any two strictly singular operators defined on

XU is compact.

◮ Every bounded linear operator defined on the space has a

non-trivial closed invariant subspace.

SLIDE 97

Saturation under constraints - new method

Properties of the space XU

◮ It is hereditarily indecomposable, it has separable dual, and it

has no boundedly complete sequences.

◮ Every Schauder basic sequence in XU admits either c0, or ℓ1, or

the summing basis of c0 as a spreading model. All three type of spreading models appear in every subspace of XU.

◮ For every bounded linear operator T defined on XU there is a

scalar λ so that S = λI − T is weakly compact and hence strictly singular.

◮ The composition of any two strictly singular operators defined on

XU is compact.

◮ Every bounded linear operator defined on the space has a

non-trivial closed invariant subspace.

SLIDE 98

Saturation under constraints - new method

Properties of the space XU

◮ It is hereditarily indecomposable, it has separable dual, and it

has no boundedly complete sequences.

◮ Every Schauder basic sequence in XU admits either c0, or ℓ1, or

the summing basis of c0 as a spreading model. All three type of spreading models appear in every subspace of XU.

◮ For every bounded linear operator T defined on XU there is a

scalar λ so that S = λI − T is weakly compact and hence strictly singular.

◮ The composition of any two strictly singular operators defined on

XU is compact.

◮ Every bounded linear operator defined on the space has a

non-trivial closed invariant subspace.

SLIDE 99

Saturation under constraints - new method

Properties of the space XU

◮ It is hereditarily indecomposable, it has separable dual, and it

has no boundedly complete sequences.

◮ Every Schauder basic sequence in XU admits either c0, or ℓ1, or

the summing basis of c0 as a spreading model. All three type of spreading models appear in every subspace of XU.

◮ For every bounded linear operator T defined on XU there is a

scalar λ so that S = λI − T is weakly compact and hence strictly singular.

◮ The composition of any two strictly singular operators defined on

XU is compact.

◮ Every bounded linear operator defined on the space has a

non-trivial closed invariant subspace.

SLIDE 100

Saturation under constraints - new method

Properties of the space XU

◮ It is hereditarily indecomposable, it has separable dual, and it

has no boundedly complete sequences.

◮ Every Schauder basic sequence in XU admits either c0, or ℓ1, or

the summing basis of c0 as a spreading model. All three type of spreading models appear in every subspace of XU.

◮ For every bounded linear operator T defined on XU there is a

scalar λ so that S = λI − T is weakly compact and hence strictly singular.

◮ The composition of any two strictly singular operators defined on

XU is compact.

◮ Every bounded linear operator defined on the space has a

non-trivial closed invariant subspace.

SLIDE 101

Saturation under constraints - new method

Properties of the space XU

◮ It is hereditarily indecomposable, it has separable dual, and it

has no boundedly complete sequences.

◮ Every Schauder basic sequence in XU admits either c0, or ℓ1, or

the summing basis of c0 as a spreading model. All three type of spreading models appear in every subspace of XU.

◮ For every bounded linear operator T defined on XU there is a

scalar λ so that S = λI − T is weakly compact and hence strictly singular.

◮ The composition of any two strictly singular operators defined on

XU is compact.

◮ Every bounded linear operator defined on the space has a

non-trivial closed invariant subspace.

SLIDE 102

Saturation under constraints - new method

It is necessary to determine spreading models generated by

normalized block sequence in XU

This is done using the α-index, an index evaluating the action of

αc-averages on the elements of a sequence.

When the α-index of a normalized block sequence is zero, then it

admits a c0 spreading model, otherwise it admits an ℓ1-spreading model.

Connection between asymptotic structure and operators: an
perator T : XU → XU is not strictly singular, if there exists a block

sequence (xk)k generating the same spreading model as (Txk)k.

It quickly follows that the composition of any two strictly singular
perators is compact.

SLIDE 103

Saturation under constraints - new method

It is necessary to determine spreading models generated by

normalized block sequence in XU

This is done using the α-index, an index evaluating the action of

αc-averages on the elements of a sequence.

When the α-index of a normalized block sequence is zero, then it

admits a c0 spreading model, otherwise it admits an ℓ1-spreading model.

Connection between asymptotic structure and operators: an
perator T : XU → XU is not strictly singular, if there exists a block

sequence (xk)k generating the same spreading model as (Txk)k.

It quickly follows that the composition of any two strictly singular
perators is compact.

SLIDE 104

Saturation under constraints - new method

It is necessary to determine spreading models generated by

normalized block sequence in XU

This is done using the α-index, an index evaluating the action of

αc-averages on the elements of a sequence.

When the α-index of a normalized block sequence is zero, then it

admits a c0 spreading model, otherwise it admits an ℓ1-spreading model.

Connection between asymptotic structure and operators: an
perator T : XU → XU is not strictly singular, if there exists a block

sequence (xk)k generating the same spreading model as (Txk)k.

It quickly follows that the composition of any two strictly singular
perators is compact.

SLIDE 105

Saturation under constraints - new method

It is necessary to determine spreading models generated by

normalized block sequence in XU

This is done using the α-index, an index evaluating the action of

αc-averages on the elements of a sequence.

When the α-index of a normalized block sequence is zero, then it

admits a c0 spreading model, otherwise it admits an ℓ1-spreading model.

Connection between asymptotic structure and operators: an
perator T : XU → XU is not strictly singular, if there exists a block

sequence (xk)k generating the same spreading model as (Txk)k.

It quickly follows that the composition of any two strictly singular
perators is compact.

SLIDE 106

Saturation under constraints - new method

It is necessary to determine spreading models generated by

normalized block sequence in XU

This is done using the α-index, an index evaluating the action of

αc-averages on the elements of a sequence.

When the α-index of a normalized block sequence is zero, then it

admits a c0 spreading model, otherwise it admits an ℓ1-spreading model.

Connection between asymptotic structure and operators: an
perator T : XU → XU is not strictly singular, if there exists a block

sequence (xk)k generating the same spreading model as (Txk)k.

It quickly follows that the composition of any two strictly singular
perators is compact.

SLIDE 107

Saturation under constraints - new method

To show that the space contains no boundedly complete

sequences, given a normalized block sequence (yk)k we find a further bounded block sequence (xk) and fk in W so that

{(fk, xk)n

k=1}∞ n=1 is a maximal chain in U and fk(xm) = δk.m.

Then, (xk)k is seminormilzed and supn n

k=1 xk < ∞.

Remark: The building blocks used to find (xk)k, are vectors of a sequence generating a c0 spreading model.

SLIDE 108

Saturation under constraints - new method

To show that the space contains no boundedly complete

sequences, given a normalized block sequence (yk)k we find a further bounded block sequence (xk) and fk in W so that

{(fk, xk)n

k=1}∞ n=1 is a maximal chain in U and fk(xm) = δk.m.

Then, (xk)k is seminormilzed and supn n

k=1 xk < ∞.

Remark: The building blocks used to find (xk)k, are vectors of a sequence generating a c0 spreading model.

SLIDE 109

Saturation under constraints - new method

To show that the space contains no boundedly complete

sequences, given a normalized block sequence (yk)k we find a further bounded block sequence (xk) and fk in W so that

{(fk, xk)n

k=1}∞ n=1 is a maximal chain in U and fk(xm) = δk.m.

Then, (xk)k is seminormilzed and supn n

k=1 xk < ∞.

Remark: The building blocks used to find (xk)k, are vectors of a sequence generating a c0 spreading model.

SLIDE 110

Saturation under constraints - new method

To show that the space contains no boundedly complete

sequences, given a normalized block sequence (yk)k we find a further bounded block sequence (xk) and fk in W so that

{(fk, xk)n

k=1}∞ n=1 is a maximal chain in U and fk(xm) = δk.m.

Then, (xk)k is seminormilzed and supn n

k=1 xk < ∞.

Remark: The building blocks used to find (xk)k, are vectors of a sequence generating a c0 spreading model.

SLIDE 111

Bourgain-Delbaen L∞-spaces

The previously mentioned L∞ HI space Xnr is constructed

combining the new variation of saturation under constraints with the Bourgain-Delbaen construction method.

A Bourgain-Delbaen L∞ space is a separable subspace X(Γq,iq)q of

ℓ∞(Γ), where Γ is a countable set, defined as follows:

There exists an increasing sequence (Γq)q of finite subsets of Γ,

such that Γ = ∪qΓq.

There exists a sequence (iq)q with each iq : ℓ∞(Γq) → ℓ∞(Γ) being

an extension linear operator (i.e. rq(iq(x)) = x, where rq : ℓ∞(Γ) → ℓ∞(Γq) the restriction operator) and (iq)q is uniformly bounded.

The sequence (iq)q is compatible, that is

ip = iq ◦ rq ◦ ip, for all p < q.

SLIDE 112

Bourgain-Delbaen L∞-spaces

The previously mentioned L∞ HI space Xnr is constructed

combining the new variation of saturation under constraints with the Bourgain-Delbaen construction method.

A Bourgain-Delbaen L∞ space is a separable subspace X(Γq,iq)q of

ℓ∞(Γ), where Γ is a countable set, defined as follows:

There exists an increasing sequence (Γq)q of finite subsets of Γ,

such that Γ = ∪qΓq.

There exists a sequence (iq)q with each iq : ℓ∞(Γq) → ℓ∞(Γ) being

an extension linear operator (i.e. rq(iq(x)) = x, where rq : ℓ∞(Γ) → ℓ∞(Γq) the restriction operator) and (iq)q is uniformly bounded.

The sequence (iq)q is compatible, that is

ip = iq ◦ rq ◦ ip, for all p < q.

SLIDE 113

Bourgain-Delbaen L∞-spaces

The previously mentioned L∞ HI space Xnr is constructed

combining the new variation of saturation under constraints with the Bourgain-Delbaen construction method.

A Bourgain-Delbaen L∞ space is a separable subspace X(Γq,iq)q of

ℓ∞(Γ), where Γ is a countable set, defined as follows:

There exists an increasing sequence (Γq)q of finite subsets of Γ,

such that Γ = ∪qΓq.

There exists a sequence (iq)q with each iq : ℓ∞(Γq) → ℓ∞(Γ) being

an extension linear operator (i.e. rq(iq(x)) = x, where rq : ℓ∞(Γ) → ℓ∞(Γq) the restriction operator) and (iq)q is uniformly bounded.

The sequence (iq)q is compatible, that is

ip = iq ◦ rq ◦ ip, for all p < q.

SLIDE 114

Bourgain-Delbaen L∞-spaces

The previously mentioned L∞ HI space Xnr is constructed

combining the new variation of saturation under constraints with the Bourgain-Delbaen construction method.

A Bourgain-Delbaen L∞ space is a separable subspace X(Γq,iq)q of

ℓ∞(Γ), where Γ is a countable set, defined as follows:

There exists an increasing sequence (Γq)q of finite subsets of Γ,

such that Γ = ∪qΓq.

There exists a sequence (iq)q with each iq : ℓ∞(Γq) → ℓ∞(Γ) being

an extension linear operator (i.e. rq(iq(x)) = x, where rq : ℓ∞(Γ) → ℓ∞(Γq) the restriction operator) and (iq)q is uniformly bounded.

The sequence (iq)q is compatible, that is

ip = iq ◦ rq ◦ ip, for all p < q.

SLIDE 115

Bourgain-Delbaen L∞-spaces

The previously mentioned L∞ HI space Xnr is constructed

combining the new variation of saturation under constraints with the Bourgain-Delbaen construction method.

A Bourgain-Delbaen L∞ space is a separable subspace X(Γq,iq)q of

ℓ∞(Γ), where Γ is a countable set, defined as follows:

There exists an increasing sequence (Γq)q of finite subsets of Γ,

such that Γ = ∪qΓq.

There exists a sequence (iq)q with each iq : ℓ∞(Γq) → ℓ∞(Γ) being

an extension linear operator (i.e. rq(iq(x)) = x, where rq : ℓ∞(Γ) → ℓ∞(Γq) the restriction operator) and (iq)q is uniformly bounded.

The sequence (iq)q is compatible, that is

ip = iq ◦ rq ◦ ip, for all p < q.

SLIDE 116

Bourgain-Delbaen L∞-spaces

The previously mentioned L∞ HI space Xnr is constructed

combining the new variation of saturation under constraints with the Bourgain-Delbaen construction method.

A Bourgain-Delbaen L∞ space is a separable subspace X(Γq,iq)q of

ℓ∞(Γ), where Γ is a countable set, defined as follows:

There exists an increasing sequence (Γq)q of finite subsets of Γ,

such that Γ = ∪qΓq.

There exists a sequence (iq)q with each iq : ℓ∞(Γq) → ℓ∞(Γ) being

an extension linear operator (i.e. rq(iq(x)) = x, where rq : ℓ∞(Γ) → ℓ∞(Γq) the restriction operator) and (iq)q is uniformly bounded.

The sequence (iq)q is compatible, that is

ip = iq ◦ rq ◦ ip, for all p < q.

SLIDE 117

Bourgain Delbaen L∞-spaces

We set ∆1 = Γ1 and for q ∈ N, ∆q+1 = Γq+1 \ Γq.
For γ ∈ ∆q, we denote dγ = iq(eγ).
The Bourgain-Delbaen space X(Γq,iq)q determined by the sequence

(Γq, iq)q is the closed subspace of ℓ∞(Γ) generated by the family {dγ}γ∈Γ.

For every q ∈ N,

{dγ : γ ∈ Γq} = iq(ℓ∞(Γq))

c

∼ ℓ∞(Γq), hence X(Γq,iq)q is a L∞-space.

Recently, the following was proved.

Theorem (S. Argyros, I. Gasparis and M. (2015))

Every separable infinite dimensional L∞-space is isomorphic to a Bourgain-Delbaen L∞-space.

SLIDE 118

Bourgain Delbaen L∞-spaces

We set ∆1 = Γ1 and for q ∈ N, ∆q+1 = Γq+1 \ Γq.
For γ ∈ ∆q, we denote dγ = iq(eγ).
The Bourgain-Delbaen space X(Γq,iq)q determined by the sequence

(Γq, iq)q is the closed subspace of ℓ∞(Γ) generated by the family {dγ}γ∈Γ.

For every q ∈ N,

{dγ : γ ∈ Γq} = iq(ℓ∞(Γq))

c

∼ ℓ∞(Γq), hence X(Γq,iq)q is a L∞-space.

Recently, the following was proved.

Theorem (S. Argyros, I. Gasparis and M. (2015))

Every separable infinite dimensional L∞-space is isomorphic to a Bourgain-Delbaen L∞-space.

SLIDE 119

Bourgain Delbaen L∞-spaces

We set ∆1 = Γ1 and for q ∈ N, ∆q+1 = Γq+1 \ Γq.
For γ ∈ ∆q, we denote dγ = iq(eγ).
The Bourgain-Delbaen space X(Γq,iq)q determined by the sequence

(Γq, iq)q is the closed subspace of ℓ∞(Γ) generated by the family {dγ}γ∈Γ.

For every q ∈ N,

{dγ : γ ∈ Γq} = iq(ℓ∞(Γq))

c

∼ ℓ∞(Γq), hence X(Γq,iq)q is a L∞-space.

Recently, the following was proved.

Theorem (S. Argyros, I. Gasparis and M. (2015))

Every separable infinite dimensional L∞-space is isomorphic to a Bourgain-Delbaen L∞-space.

SLIDE 120

Bourgain Delbaen L∞-spaces

We set ∆1 = Γ1 and for q ∈ N, ∆q+1 = Γq+1 \ Γq.
For γ ∈ ∆q, we denote dγ = iq(eγ).
The Bourgain-Delbaen space X(Γq,iq)q determined by the sequence

(Γq, iq)q is the closed subspace of ℓ∞(Γ) generated by the family {dγ}γ∈Γ.

For every q ∈ N,

{dγ : γ ∈ Γq} = iq(ℓ∞(Γq))

c

∼ ℓ∞(Γq), hence X(Γq,iq)q is a L∞-space.

Recently, the following was proved.

Theorem (S. Argyros, I. Gasparis and M. (2015))

Every separable infinite dimensional L∞-space is isomorphic to a Bourgain-Delbaen L∞-space.

SLIDE 121

Bourgain Delbaen L∞-spaces

We set ∆1 = Γ1 and for q ∈ N, ∆q+1 = Γq+1 \ Γq.
For γ ∈ ∆q, we denote dγ = iq(eγ).
The Bourgain-Delbaen space X(Γq,iq)q determined by the sequence

(Γq, iq)q is the closed subspace of ℓ∞(Γ) generated by the family {dγ}γ∈Γ.

For every q ∈ N,

{dγ : γ ∈ Γq} = iq(ℓ∞(Γq))

c

∼ ℓ∞(Γq), hence X(Γq,iq)q is a L∞-space.

Recently, the following was proved.

Theorem (S. Argyros, I. Gasparis and M. (2015))

Every separable infinite dimensional L∞-space is isomorphic to a Bourgain-Delbaen L∞-space.

SLIDE 122

Bourgain Delbaen L∞-spaces

Some properties of X(Γq,iq)q:

◮ We set Mq = {dγ : γ ∈ ∆q}. Then (Mq)q defines a FDD of

X(Γq,iq)q and (dγ)γ∈Γ is a Schauder basis of X(Γq,iq)q.

◮ The family (d∗ γ)γ∈Γ of the biorthogonals to (dγ)γ∈Γ generate a

subspace Y of X∗

(Γq,iq)q which is isomorphic to ℓ1(Γ). ◮ As it turns out: {d∗ γ : γ ∈ Γ} = {e∗ γ : γ ∈ Γ}

For x ∈ X∗

(Γq,iq)q, x = sup{|e∗ γ(x)| : γ ∈ Γ}.

(Because X∗

(Γq,iq)q ⊂ ℓ∞(Γ))

Hence, {e∗

γ : γ ∈ Γ} is a norming set for X(Γq,iq)q.

SLIDE 123

Bourgain Delbaen L∞-spaces

Some properties of X(Γq,iq)q:

◮ We set Mq = {dγ : γ ∈ ∆q}. Then (Mq)q defines a FDD of

X(Γq,iq)q and (dγ)γ∈Γ is a Schauder basis of X(Γq,iq)q.

◮ The family (d∗ γ)γ∈Γ of the biorthogonals to (dγ)γ∈Γ generate a

subspace Y of X∗

(Γq,iq)q which is isomorphic to ℓ1(Γ). ◮ As it turns out: {d∗ γ : γ ∈ Γ} = {e∗ γ : γ ∈ Γ}

For x ∈ X∗

(Γq,iq)q, x = sup{|e∗ γ(x)| : γ ∈ Γ}.

(Because X∗

(Γq,iq)q ⊂ ℓ∞(Γ))

Hence, {e∗

γ : γ ∈ Γ} is a norming set for X(Γq,iq)q.

SLIDE 124

Bourgain Delbaen L∞-spaces

Some properties of X(Γq,iq)q:

◮ We set Mq = {dγ : γ ∈ ∆q}. Then (Mq)q defines a FDD of

X(Γq,iq)q and (dγ)γ∈Γ is a Schauder basis of X(Γq,iq)q.

◮ The family (d∗ γ)γ∈Γ of the biorthogonals to (dγ)γ∈Γ generate a

subspace Y of X∗

(Γq,iq)q which is isomorphic to ℓ1(Γ). ◮ As it turns out: {d∗ γ : γ ∈ Γ} = {e∗ γ : γ ∈ Γ}

For x ∈ X∗

(Γq,iq)q, x = sup{|e∗ γ(x)| : γ ∈ Γ}.

(Because X∗

(Γq,iq)q ⊂ ℓ∞(Γ))

Hence, {e∗

γ : γ ∈ Γ} is a norming set for X(Γq,iq)q.

SLIDE 125

Bourgain Delbaen L∞-spaces

Some properties of X(Γq,iq)q:

◮ We set Mq = {dγ : γ ∈ ∆q}. Then (Mq)q defines a FDD of

X(Γq,iq)q and (dγ)γ∈Γ is a Schauder basis of X(Γq,iq)q.

◮ The family (d∗ γ)γ∈Γ of the biorthogonals to (dγ)γ∈Γ generate a

subspace Y of X∗

(Γq,iq)q which is isomorphic to ℓ1(Γ). ◮ As it turns out: {d∗ γ : γ ∈ Γ} = {e∗ γ : γ ∈ Γ}

For x ∈ X∗

(Γq,iq)q, x = sup{|e∗ γ(x)| : γ ∈ Γ}.

(Because X∗

(Γq,iq)q ⊂ ℓ∞(Γ))

Hence, {e∗

γ : γ ∈ Γ} is a norming set for X(Γq,iq)q.

SLIDE 126

Bourgain Delbaen L∞-spaces

Some properties of X(Γq,iq)q:

◮ We set Mq = {dγ : γ ∈ ∆q}. Then (Mq)q defines a FDD of

X(Γq,iq)q and (dγ)γ∈Γ is a Schauder basis of X(Γq,iq)q.

◮ The family (d∗ γ)γ∈Γ of the biorthogonals to (dγ)γ∈Γ generate a

subspace Y of X∗

(Γq,iq)q which is isomorphic to ℓ1(Γ). ◮ As it turns out: {d∗ γ : γ ∈ Γ} = {e∗ γ : γ ∈ Γ}

For x ∈ X∗

(Γq,iq)q, x = sup{|e∗ γ(x)| : γ ∈ Γ}.

(Because X∗

(Γq,iq)q ⊂ ℓ∞(Γ))

Hence, {e∗

γ : γ ∈ Γ} is a norming set for X(Γq,iq)q.

SLIDE 127

Bourgain Delbaen L∞-spaces

Some properties of X(Γq,iq)q:

◮ We set Mq = {dγ : γ ∈ ∆q}. Then (Mq)q defines a FDD of

X(Γq,iq)q and (dγ)γ∈Γ is a Schauder basis of X(Γq,iq)q.

◮ The family (d∗ γ)γ∈Γ of the biorthogonals to (dγ)γ∈Γ generate a

subspace Y of X∗

(Γq,iq)q which is isomorphic to ℓ1(Γ). ◮ As it turns out: {d∗ γ : γ ∈ Γ} = {e∗ γ : γ ∈ Γ}

For x ∈ X∗

(Γq,iq)q, x = sup{|e∗ γ(x)| : γ ∈ Γ}.

(Because X∗

(Γq,iq)q ⊂ ℓ∞(Γ))

Hence, {e∗

◮ The space Y = {dγ : γ ∈ Γ \ Γ′} is a L∞-space. ◮ The quotient X(Γq,iq)q/Y is a L∞-space.

SLIDE 133

L∞-quotients of Bourgain Delbaen L∞-spaces

Why is this useful?
Recall: in the non-L∞-setting, the norming set WU of the space XU

is a subset of the norming set Wα of the space T(1/2n,Sn,α).

Analogously: the L∞-space Xnr is obtained as a quotient of a

Bourgain-Delbaen L∞-space BmT by selecting an appropriate self-determined subset of the norming set ¯ Γ of BmT.

SLIDE 134

L∞-quotients of Bourgain Delbaen L∞-spaces

Why is this useful?
Recall: in the non-L∞-setting, the norming set WU of the space XU

is a subset of the norming set Wα of the space T(1/2n,Sn,α).

Analogously: the L∞-space Xnr is obtained as a quotient of a

Bourgain-Delbaen L∞-space BmT by selecting an appropriate self-determined subset of the norming set ¯ Γ of BmT.

SLIDE 135

L∞-quotients of Bourgain Delbaen L∞-spaces

Why is this useful?
Recall: in the non-L∞-setting, the norming set WU of the space XU

is a subset of the norming set Wα of the space T(1/2n,Sn,α).

Analogously: the L∞-space Xnr is obtained as a quotient of a

Bourgain-Delbaen L∞-space BmT by selecting an appropriate self-determined subset of the norming set ¯ Γ of BmT.

SLIDE 136

The Bourgain-Delbaen mixed-Tsirelson L∞-space BmT

The space BmT is constructed using a sequence of pairs of natural

numbers (mj, nj)j.

If BmT ⊂ ℓ∞(¯

Γ), for each γ ∈ ¯ Γ the coordinate-functional e∗

γ admits

a decomposition e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, each b∗

i is a

convex combination of the e∗

γ, and the ξi’s are an artifact of the

Bourgain-Delbaen method.

A γ with the above representation is said to have weight 1/mj.
Important to note: respecting the restrictions imposed to k, and the

E1 < · · · < Ek,more or less, convex combinations b∗ can be taken freely to define coordinates e∗

γ

SLIDE 137

The Bourgain-Delbaen mixed-Tsirelson L∞-space BmT

The space BmT is constructed using a sequence of pairs of natural

numbers (mj, nj)j.

If BmT ⊂ ℓ∞(¯

Γ), for each γ ∈ ¯ Γ the coordinate-functional e∗

γ admits

a decomposition e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, each b∗

i is a

convex combination of the e∗

γ, and the ξi’s are an artifact of the

Bourgain-Delbaen method.

A γ with the above representation is said to have weight 1/mj.
Important to note: respecting the restrictions imposed to k, and the

E1 < · · · < Ek,more or less, convex combinations b∗ can be taken freely to define coordinates e∗

γ

SLIDE 138

The Bourgain-Delbaen mixed-Tsirelson L∞-space BmT

The space BmT is constructed using a sequence of pairs of natural

numbers (mj, nj)j.

If BmT ⊂ ℓ∞(¯

Γ), for each γ ∈ ¯ Γ the coordinate-functional e∗

γ admits

a decomposition e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, each b∗

i is a

convex combination of the e∗

γ, and the ξi’s are an artifact of the

Bourgain-Delbaen method.

A γ with the above representation is said to have weight 1/mj.
Important to note: respecting the restrictions imposed to k, and the

E1 < · · · < Ek,more or less, convex combinations b∗ can be taken freely to define coordinates e∗

γ

SLIDE 139

The Bourgain-Delbaen mixed-Tsirelson L∞-space BmT

The space BmT is constructed using a sequence of pairs of natural

numbers (mj, nj)j.

If BmT ⊂ ℓ∞(¯

Γ), for each γ ∈ ¯ Γ the coordinate-functional e∗

γ admits

a decomposition e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, each b∗

i is a

convex combination of the e∗

γ, and the ξi’s are an artifact of the

Bourgain-Delbaen method.

A γ with the above representation is said to have weight 1/mj.
Important to note: respecting the restrictions imposed to k, and the

E1 < · · · < Ek,more or less, convex combinations b∗ can be taken freely to define coordinates e∗

γ

SLIDE 140

The Bourgain-Delbaen mixed-Tsirelson L∞-space BmT

The space BmT is constructed using a sequence of pairs of natural

numbers (mj, nj)j.

If BmT ⊂ ℓ∞(¯

Γ), for each γ ∈ ¯ Γ the coordinate-functional e∗

γ admits

a decomposition e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, each b∗

i is a

convex combination of the e∗

γ, and the ξi’s are an artifact of the

Bourgain-Delbaen method.

A γ with the above representation is said to have weight 1/mj.
Important to note: respecting the restrictions imposed to k, and the

E1 < · · · < Ek,more or less, convex combinations b∗ can be taken freely to define coordinates e∗

γ

SLIDE 141

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

We use BmT ⊂ ℓ∞(¯

Γ), in particular ¯ Γ, to choose a self-determined subset Γ of ¯ Γ that will produce a L∞-space Xnr not containing c0, ℓ1,

r reflexive subspaces.
Recall BmT has a basis, which we shall denote by (¯

dγ)γ.

We define a tree U.
Denote Q the set of all finite sequences (γk, xk)n

k=1 with γk ∈ ¯

Γ and xk block vectors that are finite rational linear combinations of (¯ dγ)¯

γ,

the basis of BmT.

Using a coding function, choose a subtree U of Q so that for all

(γk, xk)n

k=1 in T , the weight of γn uniquely determines the sequence

(γk, xk)n−1

k=1.

SLIDE 142

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

We use BmT ⊂ ℓ∞(¯

Γ), in particular ¯ Γ, to choose a self-determined subset Γ of ¯ Γ that will produce a L∞-space Xnr not containing c0, ℓ1,

r reflexive subspaces.
Recall BmT has a basis, which we shall denote by (¯

dγ)γ.

We define a tree U.
Denote Q the set of all finite sequences (γk, xk)n

k=1 with γk ∈ ¯

Γ and xk block vectors that are finite rational linear combinations of (¯ dγ)¯

γ,

the basis of BmT.

Using a coding function, choose a subtree U of Q so that for all

(γk, xk)n

k=1 in T , the weight of γn uniquely determines the sequence

(γk, xk)n−1

k=1.

SLIDE 143

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

We use BmT ⊂ ℓ∞(¯

Γ), in particular ¯ Γ, to choose a self-determined subset Γ of ¯ Γ that will produce a L∞-space Xnr not containing c0, ℓ1,

r reflexive subspaces.
Recall BmT has a basis, which we shall denote by (¯

dγ)γ.

We define a tree U.
Denote Q the set of all finite sequences (γk, xk)n

k=1 with γk ∈ ¯

Γ and xk block vectors that are finite rational linear combinations of (¯ dγ)¯

γ,

the basis of BmT.

Using a coding function, choose a subtree U of Q so that for all

(γk, xk)n

k=1 in T , the weight of γn uniquely determines the sequence

(γk, xk)n−1

k=1.

SLIDE 144

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

We use BmT ⊂ ℓ∞(¯

Γ), in particular ¯ Γ, to choose a self-determined subset Γ of ¯ Γ that will produce a L∞-space Xnr not containing c0, ℓ1,

r reflexive subspaces.
Recall BmT has a basis, which we shall denote by (¯

dγ)γ.

We define a tree U.
Denote Q the set of all finite sequences (γk, xk)n

k=1 with γk ∈ ¯

Γ and xk block vectors that are finite rational linear combinations of (¯ dγ)¯

γ,

the basis of BmT.

Using a coding function, choose a subtree U of Q so that for all

(γk, xk)n

k=1 in T , the weight of γn uniquely determines the sequence

(γk, xk)n−1

k=1.

SLIDE 145

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

We use BmT ⊂ ℓ∞(¯

Γ), in particular ¯ Γ, to choose a self-determined subset Γ of ¯ Γ that will produce a L∞-space Xnr not containing c0, ℓ1,

r reflexive subspaces.
Recall BmT has a basis, which we shall denote by (¯

dγ)γ.

We define a tree U.
Denote Q the set of all finite sequences (γk, xk)n

k=1 with γk ∈ ¯

Γ and xk block vectors that are finite rational linear combinations of (¯ dγ)¯

γ,

the basis of BmT.

Using a coding function, choose a subtree U of Q so that for all

(γk, xk)n

k=1 in T , the weight of γn uniquely determines the sequence

(γk, xk)n−1

k=1.

SLIDE 146

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

The tree U determines specific allowable averages, called

αc-averages.

Those are be used instead of arbitrary convex combinations when

building coordinate functionals in Γ.

More precisely, Γ is the smallest subset of ¯

Γ so that each γ ∈ Γ has an analysis of the form e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, and each b∗

i is

an αc-average, i.e. an average allowed by the tree U.

SLIDE 147

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

The tree U determines specific allowable averages, called

αc-averages.

Those are be used instead of arbitrary convex combinations when

building coordinate functionals in Γ.

More precisely, Γ is the smallest subset of ¯

Γ so that each γ ∈ Γ has an analysis of the form e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, and each b∗

i is

an αc-average, i.e. an average allowed by the tree U.

SLIDE 148

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

The tree U determines specific allowable averages, called

αc-averages.

Those are be used instead of arbitrary convex combinations when

building coordinate functionals in Γ.

More precisely, Γ is the smallest subset of ¯

Γ so that each γ ∈ Γ has an analysis of the form e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, and each b∗

i is

an αc-average, i.e. an average allowed by the tree U.

SLIDE 149

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

The tree U determines specific allowable averages, called

αc-averages.

Those are be used instead of arbitrary convex combinations when

building coordinate functionals in Γ.

More precisely, Γ is the smallest subset of ¯

Γ so that each γ ∈ Γ has an analysis of the form e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, and each b∗

i is

an αc-average, i.e. an average allowed by the tree U.

SLIDE 150

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

The tree U determines specific allowable averages, called

αc-averages.

Those are be used instead of arbitrary convex combinations when

building coordinate functionals in Γ.

More precisely, Γ is the smallest subset of ¯

Γ so that each γ ∈ Γ has an analysis of the form e∗

γ = 1

mj

k

i=1

b∗

i ◦ PEi + k

i=1

d∗

ηi,

where k nj, E1 < · · · < Ek are intervals of N, the PEi are the canonical projections with respect to the FDD (Mq)q, and each b∗

i is

an αc-average, i.e. an average allowed by the tree U.

SLIDE 151

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

One example: if (γk, xk)n

k=1 is in U so that γk ∈ Γ, E1 < · · · < En are

intervals of N so that e∗γk ◦ PEk (xk) = e∗γm ◦ PEk (xm) for 1 k m n, then b∗ = 1 n

n

k=1

(−1)ke∗

γk

is an αc-average of Γ. Specifically, averages similar to the above are called compatible averages of Γ. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements γ1, . . . < gan
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 152

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

One example: if (γk, xk)n

k=1 is in U so that γk ∈ Γ, E1 < · · · < En are

intervals of N so that e∗γk ◦ PEk (xk) = e∗γm ◦ PEk (xm) for 1 k m n, then b∗ = 1 n

n

k=1

(−1)ke∗

γk

is an αc-average of Γ. Specifically, averages similar to the above are called compatible averages of Γ. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements γ1, . . . < gan
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 153

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

One example: if (γk, xk)n

k=1 is in U so that γk ∈ Γ, E1 < · · · < En are

intervals of N so that e∗γk ◦ PEk (xk) = e∗γm ◦ PEk (xm) for 1 k m n, then b∗ = 1 n

n

k=1

(−1)ke∗

γk

is an αc-average of Γ. Specifically, averages similar to the above are called compatible averages of Γ. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements γ1, . . . < gan
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 154

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

One example: if (γk, xk)n

k=1 is in U so that γk ∈ Γ, E1 < · · · < En are

intervals of N so that e∗γk ◦ PEk (xk) = e∗γm ◦ PEk (xm) for 1 k m n, then b∗ = 1 n

n

k=1

(−1)ke∗

γk

is an αc-average of Γ. Specifically, averages similar to the above are called compatible averages of Γ. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements γ1, . . . < gan
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 155

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

One example: if (γk, xk)n

k=1 is in U so that γk ∈ Γ, E1 < · · · < En are

intervals of N so that e∗γk ◦ PEk (xk) = e∗γm ◦ PEk (xm) for 1 k m n, then b∗ = 1 n

n

k=1

(−1)ke∗

γk

is an αc-average of Γ. Specifically, averages similar to the above are called compatible averages of Γ. Comment: this type of averages will provide HI structure to the space and not special functionals.

Other types of αc-averages if G are built on elements γ1, . . . < gan
f G that are “incomparable” or “irrelevant” with respect to U.

SLIDE 156

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

In conclusion, if Y = {¯

dγ : γ ∈ ¯ Γ \ Γ}, then Xnr is the desired space.

Properties of the space Xnr

◮ It is hereditarily indecomposable L∞-space, whose dual is

isomorphic to ℓ1, and it has no boundedly complete.

◮ In particular, Xnr is a L∞-space not containing c0, ℓ1, or reflexive

subspaces.

◮ Every bounded linear operator T : Xnr → Xnr is a scalar multiple

f the identity plus a compact operator.

SLIDE 157

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

In conclusion, if Y = {¯

dγ : γ ∈ ¯ Γ \ Γ}, then Xnr is the desired space.

Properties of the space Xnr

◮ It is hereditarily indecomposable L∞-space, whose dual is

isomorphic to ℓ1, and it has no boundedly complete.

◮ In particular, Xnr is a L∞-space not containing c0, ℓ1, or reflexive

subspaces.

◮ Every bounded linear operator T : Xnr → Xnr is a scalar multiple

f the identity plus a compact operator.

SLIDE 158

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

In conclusion, if Y = {¯

dγ : γ ∈ ¯ Γ \ Γ}, then Xnr is the desired space.

Properties of the space Xnr

◮ It is hereditarily indecomposable L∞-space, whose dual is

isomorphic to ℓ1, and it has no boundedly complete.

◮ In particular, Xnr is a L∞-space not containing c0, ℓ1, or reflexive

subspaces.

◮ Every bounded linear operator T : Xnr → Xnr is a scalar multiple

f the identity plus a compact operator.

SLIDE 159

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

In conclusion, if Y = {¯

dγ : γ ∈ ¯ Γ \ Γ}, then Xnr is the desired space.

Properties of the space Xnr

◮ It is hereditarily indecomposable L∞-space, whose dual is

isomorphic to ℓ1, and it has no boundedly complete.

◮ In particular, Xnr is a L∞-space not containing c0, ℓ1, or reflexive

subspaces.

◮ Every bounded linear operator T : Xnr → Xnr is a scalar multiple

f the identity plus a compact operator.

SLIDE 160

A mixed-Tsirelson Bourgain-Delbaen L∞-space with new saturation under constraints

In conclusion, if Y = {¯

dγ : γ ∈ ¯ Γ \ Γ}, then Xnr is the desired space.

Properties of the space Xnr

◮ It is hereditarily indecomposable L∞-space, whose dual is

isomorphic to ℓ1, and it has no boundedly complete.

◮ In particular, Xnr is a L∞-space not containing c0, ℓ1, or reflexive

subspaces.

◮ Every bounded linear operator T : Xnr → Xnr is a scalar multiple

f the identity plus a compact operator.

On HI Banach spaces without reflexive subspaces and their spaces of operator

Pavlos Motakis (joint work with S. Argyros)

Transfinite methods in Banach spaces and algebras of operators July 20, 2016 - B˛ edlewo

Contents

property without reflexive subspaces.

saturation under constraints.

General discussion - L∞-spaces

Definition (J. Lindenstrauss - A. Pełczy´ nski (1968))

If X is separable a separable Banach space then X is called a L∞-space if there exists C 1 with X = ∪nEn, where (En)n is an increasing sequence of finite dimensional spaces with each En C-isomorphic to ℓdim En

.

e.g. C[0, 1], ℓ∞(N), or c0(N).

constructed via the Bourgain-Delbaen construction method (1980), which was invented to present examples of L∞-spaces not containing c0.

General discussion - L∞-spaces

Definition (J. Lindenstrauss - A. Pełczy´ nski (1968))

If X is separable a separable Banach space then X is called a L∞-space if there exists C 1 with X = ∪nEn, where (En)n is an increasing sequence of finite dimensional spaces with each En C-isomorphic to ℓdim En

.

e.g. C[0, 1], ℓ∞(N), or c0(N).

constructed via the Bourgain-Delbaen construction method (1980), which was invented to present examples of L∞-spaces not containing c0.

General discussion - L∞-spaces

Definition (J. Lindenstrauss - A. Pełczy´ nski (1968))

If X is separable a separable Banach space then X is called a L∞-space if there exists C 1 with X = ∪nEn, where (En)n is an increasing sequence of finite dimensional spaces with each En C-isomorphic to ℓdim En

.

e.g. C[0, 1], ℓ∞(N), or c0(N).

constructed via the Bourgain-Delbaen construction method (1980), which was invented to present examples of L∞-spaces not containing c0.

General discussion - a problem under consideration

Problem (J. Bourgain (1981)

Does there exist a Banach space not containing c0, ℓ1, or reflexive subspaces?

implying that such spaces that are moreover L∞ might not exist, i.e. that every L∞-space contains either c0, or ℓ1, or reflexive subspaces.

General discussion - a problem under consideration

Problem (J. Bourgain (1981)

Does there exist a Banach space not containing c0, ℓ1, or reflexive subspaces?

implying that such spaces that are moreover L∞ might not exist, i.e. that every L∞-space contains either c0, or ℓ1, or reflexive subspaces.

General discussion - a problem under consideration

Problem (J. Bourgain (1981)

Does there exist a Banach space not containing c0, ℓ1, or reflexive subspaces?

implying that such spaces that are moreover L∞ might not exist, i.e. that every L∞-space contains either c0, or ℓ1, or reflexive subspaces.

General discussion - a problem under consideration

Problem (J. Bourgain (1981)

Does there exist a Banach space not containing c0, ℓ1, or reflexive subspaces?

implying that such spaces that are moreover L∞ might not exist, i.e. that every L∞-space contains either c0, or ℓ1, or reflexive subspaces.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

reflexive subspaces cannot contain boundedly complete

unconditional sequences.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

reflexive subspaces cannot contain boundedly complete

unconditional sequences.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

reflexive subspaces cannot contain boundedly complete

unconditional sequences.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

reflexive subspaces cannot contain boundedly complete

unconditional sequences.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

reflexive subspaces cannot contain boundedly complete

unconditional sequences.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

reflexive subspaces cannot contain boundedly complete

unconditional sequences.

Definition

A Banach space X is called decomposable if it can be written as a direct sum X = Y ⊕ Z with both Y and Z infinite dimensional. If X contains no decomposable subspaces, then it is called hereditarily indecomposable.

Theorem (C. Stegall (1973))

If X is a separable L∞-space with non-separable dual then X contains an isomorphic copy of ℓ1. Conclusion:

reflexive subspaces cannot contain boundedly complete

unconditional sequences.