[PPT] - Maximal Ideals of Triangular Operator Algebras John Lindsay Orr PowerPoint Presentation

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Maximal Ideals of Triangular Operator Algebras

John Lindsay Orr jorr@math.unl.edu

University of Nebraska – Lincoln and Lancaster University

May 17, 2007

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 1 / 41

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http://www.math.unl.edu/∼jorr/presentations

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 2 / 41

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Ideals of upper triangular operators Statement of the problem

Let H := ℓ2(N) and let {ek}∞

k=1 be the standard basis. Let T be the

algebra of all (bounded) operators which are upper triangular with respect to {ek}.

Question

What are the maximal two-sided ideals of T ?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 3 / 41

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Ideals of upper triangular operators Statement of the problem

Let H := ℓ2(N) and let {ek}∞

k=1 be the standard basis. Let T be the

algebra of all (bounded) operators which are upper triangular with respect to {ek}.

Question

What are the maximal two-sided ideals of T ? All ideals are assumed two-sided.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 3 / 41

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Ideals of upper triangular operators Statement of the problem

What would I like the answer to be?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 4 / 41

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Ideals of upper triangular operators Statement of the problem

What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {ek} is *-isomorphic to ℓ∞(N), so we identify them. Write S for the set of strictly upper triangular operators w.r.t. {ek}.

Fact

Let M be a maximal ideal of ℓ∞(N) and let J := M + S. Then J is a maximal ideal of T .

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 4 / 41

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Ideals of upper triangular operators Statement of the problem

What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {ek} is *-isomorphic to ℓ∞(N), so we identify them. Write S for the set of strictly upper triangular operators w.r.t. {ek}.

Fact

Let M be a maximal ideal of ℓ∞(N) and let J := M + S. Then J is a maximal ideal of T .

Proof.

Write ∆(T) for the diagonal part of T. Suppose T ∈ J .

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 4 / 41

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Ideals of upper triangular operators Statement of the problem

What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {ek} is *-isomorphic to ℓ∞(N), so we identify them. Write S for the set of strictly upper triangular operators w.r.t. {ek}.

Fact

Let M be a maximal ideal of ℓ∞(N) and let J := M + S. Then J is a maximal ideal of T .

Proof.

Write ∆(T) for the diagonal part of T. Suppose T ∈ J . T − ∆(T) = J ∈ S ⊆ J and so ∆(T) ∈ J , hence ∆(T) ∈ M. Thus D∆(T) + M = I and so D(T − J) + M = I ∈ T, J .

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 4 / 41

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Ideals of upper triangular operators Statement of the problem

The maximal ideals of ℓ∞(N) are points in βN, the Stone-Cech compactification of N, so this would give a good description of the maximal ideals of T .

Question

Are all the maximal ideals of T of the form M + S where M is a maximal ideal of ℓ∞(N)?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 5 / 41

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Ideals of upper triangular operators Re-statement of the problem

Proposition

TFAE:

1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S ∈ S). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 6 / 41

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Ideals of upper triangular operators Re-statement of the problem

Proposition

TFAE:

1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S ∈ S).

Proof.

(1) ⇒ (2) ⇒ (3): Obvious.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 6 / 41

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Ideals of upper triangular operators Re-statement of the problem

Proposition

TFAE:

1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S ∈ S).

Proof.

(1) ⇒ (2) ⇒ (3): Obvious. (3) ⇒ (2): Contrapositive. Suppose J ⊇ S is a maximal ideal of T . Then J + S = T and so I = J − S.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 6 / 41

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Ideals of upper triangular operators Re-statement of the problem

Proposition

TFAE:

1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S ∈ S).

Proof.

(1) ⇒ (2) ⇒ (3): Obvious. (3) ⇒ (2): Contrapositive. Suppose J ⊇ S is a maximal ideal of T . Then J + S = T and so I = J − S. (2) ⇒ (1): Let J be a maximal ideal of T . Since J ⊇ S, then also J ⊇ ∆(J ). But ∆(J ) ⊳ D so let M ⊇ ∆(J ) be a maximal ideal of D and we saw M + S is a maximal ideal of T – that contains J .

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 6 / 41

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Ideals of upper triangular operators Re-statement of the problem

Proposition

TFAE:

1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S ∈ S).

Proof.

(1) ⇒ (2) ⇒ (3): Obvious. (3) ⇒ (2): Contrapositive. Suppose J ⊇ S is a maximal ideal of T . Then J + S = T and so I = J − S. (2) ⇒ (1): Let J be a maximal ideal of T . Since J ⊇ S, then also J ⊇ ∆(J ). But ∆(J ) ⊳ D so let M ⊇ ∆(J ) be a maximal ideal of D and we saw M + S is a maximal ideal of T – that contains J .

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 6 / 41

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Ideals of upper triangular operators Re-statement of the problem

Question

Is it possible for an operator of the form I + S (S strictly upper triangular) to lie in a proper ideal of T ?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 7 / 41

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Ideals of upper triangular operators Re-statement of the problem

Question

Is it possible for an operator of the form I + S (S strictly upper triangular) to lie in a proper ideal of T ? Just to be clear, an operator X fails to belong to a proper ideal of T iff we can find A1, . . . , An and B1, . . . , Bn such that A1XB1 + · · · + AnXBn = I

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 7 / 41

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Ideals of upper triangular operators Operators of the form I + S

In finite dimensions, all operators I + S are invertible.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 8 / 41

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Ideals of upper triangular operators Operators of the form I + S

In finite dimensions, all operators I + S are invertible. Not so in infinite dimensions. Let      1 1 1 ... ... ...      be the unilateral backward shift Then I − U =      1 −1 1 −1 1 −1 ... ... ... ...      is not invertible

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 8 / 41

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Ideals of upper triangular operators Operators of the form I + S

Nevertheless this isn’t a counterexample. It’s easy to see that I − U doesn’t lie in any proper ideal of T :

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 9 / 41

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Ideals of upper triangular operators Operators of the form I + S

Nevertheless this isn’t a counterexample. It’s easy to see that I − U doesn’t lie in any proper ideal of T : Let σ ⊆ N and let Pσ := Proj (span{ek : k ∈ σ}) Note UP2N = P2N−1U and UP2N−1 = P2NU

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 9 / 41

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Ideals of upper triangular operators Operators of the form I + S

Nevertheless this isn’t a counterexample. It’s easy to see that I − U doesn’t lie in any proper ideal of T : Let σ ⊆ N and let Pσ := Proj (span{ek : k ∈ σ}) Note UP2N = P2N−1U and UP2N−1 = P2NU Thus P2N(I − U)P2N + P2N−1(I − U)P2N−1 = I

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 9 / 41

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Ideals of upper triangular operators Operators of the form I + S

Nevertheless this isn’t a counterexample. It’s easy to see that I − U doesn’t lie in any proper ideal of T : Let σ ⊆ N and let Pσ := Proj (span{ek : k ∈ σ}) Note UP2N = P2N−1U and UP2N−1 = P2NU Thus P2N(I − U)P2N + P2N−1(I − U)P2N−1 = I This simple observation connects us to a famous open problem known as The Kadison-Singer problem or The Paving Problem.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 9 / 41

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The Kadison-Singer Problem The paving problem

Let the standard atomic masa, D, and the projections, Pσ, be as defined before.

Definition

Say that X ∈ B(H) can be “paved” if, given any ǫ > 0, there are pwd sets σ1, . . . σn ⊆ N such that σ1 ∪ · · · ∪ σn = N and

∆(X) −

n

k=1

PσkXPσk

< ǫ

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 10 / 41

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The Kadison-Singer Problem The paving problem

Let the standard atomic masa, D, and the projections, Pσ, be as defined before.

Definition

Say that X ∈ B(H) can be “paved” if, given any ǫ > 0, there are pwd sets σ1, . . . σn ⊆ N such that σ1 ∪ · · · ∪ σn = N and

∆(X) −

n

k=1

PσkXPσk

< ǫ

Question (Paving Problem)

Can every operator in B(H) be paved?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 10 / 41

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The Kadison-Singer Problem Application to maximal ideals of T

Proposition

If every operator can be paved, then no operator of the form I + S (S ∈ S) can belong to a proper ideal of T .

Proof.

I + S can be paved by projections in D. So

I −

n

k=1

Pσi(I + S)Pσi

< 1

and n

k=1 Pσi(I + S)Pσi is invertible in T .

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 11 / 41

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The Kadison-Singer Problem Extensions of pure states

In [KS59] Kadison and Singer studied “Extensions of Pure States”. Let B ⊆ A be C∗ algebras. If φ is a pure state of B then it extends to a state on A. Are such extensions unique?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 12 / 41

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The Kadison-Singer Problem Extensions of pure states

In [KS59] Kadison and Singer studied “Extensions of Pure States”. Let B ⊆ A be C∗ algebras. If φ is a pure state of B then it extends to a state on A. Are such extensions unique?

Question (Kadison-Singer)

Let D be an atomic masa in B(H). Does every pure state of D have a unique extension to a state of B(H)?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 12 / 41

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The Kadison-Singer Problem Extensions of pure states

If M is a non-atomic masa in B(H) (i.e. L∞(0, 1)) then it has pure states with non-unique extensions [KS59]. (In fact no pure states on L∞(0, 1) extend uniquely [And79a].) If D is an atomic masa in B(H) (i.e. ℓ∞(N)) and φ is a pure state on D, then φ · ∆ is a state on B(H). (Anderson [And79b] showed it is a pure state.) Is φ · ∆ the only extension of φ to a state of B(H)?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 13 / 41

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The Kadison-Singer Problem Extensions of pure states

Proposition

TFAE

1 Every operator in B(H) can be paved. 2 Every pure state of D has a unique state extension to B(H). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 14 / 41

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The Kadison-Singer Problem Extensions of pure states

Proposition

TFAE

1 Every operator in B(H) can be paved. 2 Every pure state of D has a unique state extension to B(H).

Proof.

(1) ⇒ (2): Let ˆ φ be a state extension of φ. Then ˆ φ is a D-bimodule

map. Thus by paving X we can arrange

φ · ∆(X) = ˆ φ · ∆(X) ∼ǫ ˆ φ n

k=1

PσiXPσi

=

n

k=1

φ(Pσi)2 ˆ φ(X) = ˆ φ(X)

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 14 / 41

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The Kadison-Singer Problem Extensions of pure states

Proposition

TFAE

1 Every operator in B(H) can be paved. 2 Every pure state of D has a unique state extension to B(H).

Proof.

(1) ⇒ (2): Let ˆ φ be a state extension of φ. Then ˆ φ is a D-bimodule

map. Thus by paving X we can arrange

φ · ∆(X) = ˆ φ · ∆(X) ∼ǫ ˆ φ n

k=1

PσiXPσi

=

n

k=1

φ(Pσi)2 ˆ φ(X) = ˆ φ(X)

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 14 / 41

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The Kadison-Singer Problem Extensions of pure states

Lemma

ˆ φ is a D-bimodule map.

Proof.

Let p ∈ D be a projection. Then ˆ φ(p) = φ(p) = φ(p)2 = 0, 1. If φ(p) = 0 then by Cauchy-Schwartz, ˆ φ(px) = 0 = ˆ φ(p)ˆ φ(x) If φ(p) = 1 then, again by Cauchy-Schwartz, ˆ φ(px) = ˆ φ(x) − ˆ φ(p⊥x) = ˆ φ(x) = ˆ φ(p)ˆ φ(x) (Extend to arbitrary a ∈ D by spectral theory.)

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 15 / 41

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The Kadison-Singer Problem Progress on the problem

Reid; [Rei71] Anderson; [And79a, And79b] Berman, Halpern, Kaftal, Weiss; [BHKW88] Bourgain, Tzafriri; [BT91] Weaver; [Wea04, Wea03] Casazza, Christensen, Lindner, Vershynin; [CCLV05] Casazza, Tremain “The paving conjecture is equivalent to the paving conjecture for triangular matrices”; [CT]

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 16 / 41

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More upper triangular ideals One-term interpolation

Return to X = I + S ∈ T (S ∈ S).

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41

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More upper triangular ideals One-term interpolation

Return to X = I + S ∈ T (S ∈ S). We want to find Ai, Bi such that A1XB1 + · · · + AnXBn = I.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41

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More upper triangular ideals One-term interpolation

Return to X = I + S ∈ T (S ∈ S). We want to find Ai, Bi such that A1XB1 + · · · + AnXBn = I. How about solving AXB = I for A, B ∈ T ?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41

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More upper triangular ideals One-term interpolation

Return to X = I + S ∈ T (S ∈ S). We want to find Ai, Bi such that A1XB1 + · · · + AnXBn = I. How about solving AXB = I for A, B ∈ T ? Unfortunately. . .

Proposition

Let X ∈ T . There are A, B ∈ T with AXB = I iff X is an invertible

perator.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41

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More upper triangular ideals One-term interpolation

Return to X = I + S ∈ T (S ∈ S). We want to find Ai, Bi such that A1XB1 + · · · + AnXBn = I. How about solving AXB = I for A, B ∈ T ? Unfortunately. . .

Proposition

Let X ∈ T . There are A, B ∈ T with AXB = I iff X is an invertible

perator.

Proof.

If AXB = I let Pn := P{1,...,n} and note Pn = (PnAPn) (PnXPn) (PnBPn) = (PnBAPn) PnXPn since PnBPn is the (two-sided) inverse of PnAXPn in PnH. Taking WOT-limits we see BAX = I and similarly XBA = I.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41

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More upper triangular ideals One-term interpolation

Return to X = I + S ∈ T (S ∈ S). We want to find Ai, Bi such that A1XB1 + · · · + AnXBn = I. How about solving AXB = I for A, B ∈ T ? Unfortunately. . .

Proposition

Let X ∈ T . There are A, B ∈ T with AXB = I iff X is an invertible

perator.

So how about solving AXB + CXD = I?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41

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More upper triangular ideals Two-term interpolation

First express as a finite dimensional problem:

Question

Given an n × n matrix X = I + S (S strictly upper triangular), can we find upper triangular matrices A, . . . , D such that AXB + CXD = I

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 18 / 41

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More upper triangular ideals Two-term interpolation

First express as a finite dimensional problem:

Question

Given an n × n matrix X = I + S (S strictly upper triangular), can we find upper triangular matrices A, . . . , D such that AXB + CXD = I where the max{A, . . . , D} is bounded in terms of X but independently of n?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 18 / 41

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More upper triangular ideals Two-term interpolation

Lemma

Let X = I + S ∈ Mn(C) where S is strictly upper triangular. Then there are A, . . . , D ∈ Mn(C) such that AXB + CXD = I and max{A, . . . , D} ≤ X.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41

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More upper triangular ideals Two-term interpolation

Lemma

Let X = I + S ∈ Mn(C) where S is strictly upper triangular. Then there are A, . . . , D ∈ Mn(C) such that AXB + CXD = I and max{A, . . . , D} ≤ X.

Proof.

Assume for simplicity n is even. Let s1 ≥ s2 ≥ · · · ≥ sn be the singular values of X.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41

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More upper triangular ideals Two-term interpolation

Lemma

Let X = I + S ∈ Mn(C) where S is strictly upper triangular. Then there are A, . . . , D ∈ Mn(C) such that AXB + CXD = I and max{A, . . . , D} ≤ X.

Proof.

Assume for simplicity n is even. Let s1 ≥ s2 ≥ · · · ≥ sn be the singular values of X. Since all si ≤ X and n

i=1 si = det |X| = | det X| = 1, we

cannot have n/2 of the si satisfying si < 1/X.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41

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More upper triangular ideals Two-term interpolation

Lemma

Let X = I + S ∈ Mn(C) where S is strictly upper triangular. Then there are A, . . . , D ∈ Mn(C) such that AXB + CXD = I and max{A, . . . , D} ≤ X.

Proof.

Assume for simplicity n is even. Let s1 ≥ s2 ≥ · · · ≥ sn be the singular values of X. Since all si ≤ X and n

i=1 si = det |X| = | det X| = 1, we

cannot have n/2 of the si satisfying si < 1/X. For in that case 1 = det X < Xn/2/Xn/2 ≤ 1. Thus the first n/2 of the si are at least X−1.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41

SLIDE 46

More upper triangular ideals Two-term interpolation

Lemma

Let X = I + S ∈ Mn(C) where S is strictly upper triangular. Then there are A, . . . , D ∈ Mn(C) such that AXB + CXD = I and max{A, . . . , D} ≤ X.

Proof.

Assume for simplicity n is even. Let s1 ≥ s2 ≥ · · · ≥ sn be the singular values of X. Since all si ≤ X and n

i=1 si = det |X| = | det X| = 1, we

cannot have n/2 of the si satisfying si < 1/X. For in that case 1 = det X < Xn/2/Xn/2 ≤ 1. Thus the first n/2 of the si are at least X−1. There are o.n. bases ui, vi(1 ≤ i ≤ n) such that Xui = sivi. Let A, B be matrices mapping vi → (1/si)ei and ei → ui for 1 ≤ i ≤ n/2. Then AXB is the projection

nto span{e1, . . . e n

2 } and A, B ≤ s−1 n 2

≤ X.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41

SLIDE 47

More upper triangular ideals Two-term interpolation

Lemma

Let X = I + S ∈ Mn(C) where S is strictly upper triangular. Then there are A, . . . , D ∈ Mn(C) such that AXB + CXD = I and max{A, . . . , D} ≤ X.

Proof.

Assume for simplicity n is even. Let s1 ≥ s2 ≥ · · · ≥ sn be the singular values of X. Since all si ≤ X and n

i=1 si = det |X| = | det X| = 1, we

cannot have n/2 of the si satisfying si < 1/X. For in that case 1 = det X < Xn/2/Xn/2 ≤ 1. Thus the first n/2 of the si are at least X−1. There are o.n. bases ui, vi(1 ≤ i ≤ n) such that Xui = sivi. Let A, B be matrices mapping vi → (1/si)ei and ei → ui for 1 ≤ i ≤ n/2. Then AXB is the projection

nto span{e1, . . . e n

2 } and A, B ≤ s−1 n 2

≤ X. Likewise get CXD as the projection onto span{e n

2 +1, . . . en} with norm control. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41

SLIDE 48

More upper triangular ideals Two-term interpolation

But – although we used the fact X is upper triangular – we lost all control

n triangularity of A, . . . , D.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 20 / 41

SLIDE 49

More upper triangular ideals Two-term interpolation

But – although we used the fact X is upper triangular – we lost all control

n triangularity of A, . . . , D.

At least we see there is no spectral obstruction to a two-term

decomposition. Might there be other obstructions? Index perhaps?

Question

Given X = I + S (S ∈ S), are there A, . . . , D ∈ T such that AXB + CXD = I?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 20 / 41

SLIDE 50

More upper triangular ideals Filters

Suppose now that there is a maximal ideal J of T that contains X = I + S (S ∈ S) and deduce some consequences. Let Σ = {σ ⊆ N : I − Pσ ∈ J }

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 21 / 41

SLIDE 51

More upper triangular ideals Filters

Proposition

Let Σ = {σ ⊆ N : I − Pσ ∈ J } Then

1 Σ is a filter. 2 Σ contains all cofinite subset of N. 3 σ ∈ Σ ⇒ σ + 1 ∈ Σ. 4 Σ is not an ultrafilter. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41

SLIDE 52

More upper triangular ideals Filters

Proposition

Let Σ = {σ ⊆ N : I − Pσ ∈ J } Then

1 Σ is a filter. 2 Σ contains all cofinite subset of N. 3 σ ∈ Σ ⇒ σ + 1 ∈ Σ. 4 Σ is not an ultrafilter.

Proof.

If σ ∈ Σ and τ ⊇ σ then Pτ c = Pτ cPσc ∈ J . If σ1, σ2 ∈ Σ then P⊥

σ1∩σ2 = Pσc

1∪σc 2 = Pσc 1 + Pσc 2 − Pσc 1Pσc 2. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41

SLIDE 53

More upper triangular ideals Filters

Proposition

Let Σ = {σ ⊆ N : I − Pσ ∈ J } Then

1 Σ is a filter. 2 Σ contains all cofinite subset of N. 3 σ ∈ Σ ⇒ σ + 1 ∈ Σ. 4 Σ is not an ultrafilter.

Proof.

For each k, P{k} = P{k}XP{k} ∈ J so {k}c ∈ Σ, a filter.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41

SLIDE 54

More upper triangular ideals Filters

Proposition

Let Σ = {σ ⊆ N : I − Pσ ∈ J } Then

1 Σ is a filter. 2 Σ contains all cofinite subset of N. 3 σ ∈ Σ ⇒ σ + 1 ∈ Σ. 4 Σ is not an ultrafilter.

Proof.

J ⊇ S and so S + J = T . Let U be the backward shift. Then UT = T U = S and so U is invertible (mod)J . But UPσ+1 = PσU so Pσ = I(mod)J iff Pσ+1 = I(mod)J .

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41

SLIDE 55

More upper triangular ideals Filters

Proposition

Let Σ = {σ ⊆ N : I − Pσ ∈ J } Then

1 Σ is a filter. 2 Σ contains all cofinite subset of N. 3 σ ∈ Σ ⇒ σ + 1 ∈ Σ. 4 Σ is not an ultrafilter.

Proof.

Neither 2N nor 2N − 1 can be in Σ for then its complement is in Σ also.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41

SLIDE 56

More upper triangular ideals Filters

Proposition

Let Σ = {σ ⊆ N : I − Pσ ∈ J } Then

1 Σ is a filter. 2 Σ contains all cofinite subset of N. 3 σ ∈ Σ ⇒ σ + 1 ∈ Σ. 4 Σ is not an ultrafilter. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41

SLIDE 57

Nest Algebras Definitions

Nest algebras

Definition (Ringrose, [Rin65])

Let H be a Hilbert space and N a complete chain of subspaces containing 0 and H. This is called a nest. Define the nest algebra, Alg(N), for a given nest N to be Alg(N) := {X ∈ B(H) : XN ⊆ N ∀N ∈ N} See Davidson, Nest Algebras, [Dav88].

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 23 / 41

SLIDE 58

Nest Algebras Definitions

Nest algebras

Definition (Ringrose, [Rin65])

Let H be a Hilbert space and N a complete chain of subspaces containing 0 and H. This is called a nest. Define the nest algebra, Alg(N), for a given nest N to be Alg(N) := {X ∈ B(H) : XN ⊆ N ∀N ∈ N} See Davidson, Nest Algebras, [Dav88].

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 23 / 41

SLIDE 59

Nest Algebras Examples

Example

Let e1, . . . , en be the standard basis for Cn. Let Ni := span{e1, . . . , ei} and N := {0, Ni : 1 ≤ i ≤ n}. Then Alg(N) = Tn(C).

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 24 / 41

SLIDE 60

Nest Algebras Examples

Example

Let ei (i ∈ N) be the standard basis for H = ℓ2(N). Let Ni := span{e1, . . . , ei} and N := {0, Ni, H : i ∈ N}. Then Alg(N) is the algebra of all bounded operators which are upper triangular w.r.t. {ei}. In other words, Alg(N) = T

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 25 / 41

SLIDE 61

Nest Algebras Examples

The Volterra Nest

Example

Let H = L2(0, 1). For each t ∈ [0, 1] let Nt := {f ∈ L2(0, 1) : f is supported a.e. on [0,t]} In other words, P(Nt) is multiplication by χ[0,t]. Clearly N := {Nt : t ∈ [0, 1]} is a nest.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 26 / 41

SLIDE 62

Nest Algebras Examples

The Volterra Nest

Example

Let H = L2(0, 1). For each t ∈ [0, 1] let Nt := {f ∈ L2(0, 1) : f is supported a.e. on [0,t]} In other words, P(Nt) is multiplication by χ[0,t]. Clearly N := {Nt : t ∈ [0, 1]} is a nest.

Remark

Alg(N) contains the Volterra integral operator, f − → 1

x

f (t) dt

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 26 / 41

SLIDE 63

Nest Algebras Classification and similarity

Classification of nest algebras

Theorem (Ringrose, [Rin66])

Let φ : Alg(N1) → Alg(N2) be an algebraic isomorphsim. Then there is an invertible operator S ∈ B(H1, H2) such that φ(T) = STS−1 = AdS(T) for all T ∈ Alg(N1) Now φ = AdS iff {SN : N ∈ N1} = N2. So classifying nest algebras up to isomorphism means classifying nests up to similarity.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 27 / 41

SLIDE 64

Nest Algebras Classification and similarity

Theorem (Erdos, [Erd67])

Nests are completely classified up to unitary equivalence by An order type A measure class, and A multiplicity function C.f. Unitary invariants for bounded selfadjoint operators (spectrum, measure class, mutliplicity function).

Question

Any similarity transform preserves order type. Must it also preserve multiplicity and/or measure class?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 28 / 41

SLIDE 65

Nest Algebras Classification and similarity

Let N be the Volterra nest on H = L2(0, 1). I.e. N = {Nt : t ∈ [0, 1]} where Nt = {f : f (x) = 0 a.e. x ∈ [0, t]}

Example

The map Nt − → Nt ⊕ Nt preserves order type and measure class, but not spectral multiplicity.

Example

Let f : [0, 1] → [0, 1] be increasing, bjijective, not absolutely continuous. The map Nt − → Nf (t) preserves order type and multiplicity, but not measure class.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 29 / 41

SLIDE 66

Nest Algebras Classification and similarity

Theorem (Davidson, [Dav84])

Let N1, N2 be nests and θ : N1 → N2 be and order isomorphism. There is an invertible operator S such that θ(N) = SN for all N ∈ N1 iff θ is dimension-preserving, i.e. if dim θ(N) ⊖ θ(M) = dim N ⊖ M for all M < N in N1

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 30 / 41

SLIDE 67

Nest Algebras Classification and similarity

Corollary

Both of the previous two examples are implemented by invertibles!

Corollary

Nest algebras are classified up to isomorphism by “order-dimension” type. Proof uses Voiculescu’s notion of approximate unitary equivalence. Based on N. T. Andersen’s study of unitary equivalence of quasi-triangular algebras Slightly earlier result of D. Larson [Lar85] showed all continuous nests are similar.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 31 / 41

SLIDE 68

Nest Algebras Algebraic implications of Similarity Theory

Proposition

The commutator ideal of a continuous nest is the whole algebra.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 32 / 41

SLIDE 69

Nest Algebras Algebraic implications of Similarity Theory

Proposition

The commutator ideal of a continuous nest is the whole algebra.

Proof.

By the Similarity Theorem, Alg(N) ∼ = Alg(N ⊕ N) = M2(Alg(N)) and so 1 1 1

−

1 1 1 2 = I

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 32 / 41

SLIDE 70

Nest Algebras Algebraic implications of Similarity Theory

Corollary

Let N be the Volterra nest. Then there is no ideal S ⊳ Alg(N) such that Alg(N) = D(N) ⊕ S.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 33 / 41

SLIDE 71

Nest Algebras Algebraic implications of Similarity Theory

Corollary

Let N be the Volterra nest. Then there is no ideal S ⊳ Alg(N) such that Alg(N) = D(N) ⊕ S.

Proof.

D(N) = N ′ = N ′′ is abelian so S would contain the commutator ideal.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 33 / 41

SLIDE 72

Nest Algebras Algebraic implications of Similarity Theory

Proposition

Alg(N) has non-zero idempotents which are “zero on the diagonal”, i.e. P(Nbi − Nai) Q P(Nbi − Nai) = 0 where

i

P(Nbi − Nai) = I

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 34 / 41

SLIDE 73

Nest Algebras Algebraic implications of Similarity Theory

Proposition

Alg(N) has non-zero idempotents which are “zero on the diagonal”, i.e. P(Nbi − Nai) Q P(Nbi − Nai) = 0 where

i

P(Nbi − Nai) = I

Proof.

Write the Cantor middle- 1

3 set as K = [0, 1] \ ∞ i=1(ai, bi). Let

f : [0, 1] → [0, 1] map K to a non-null set. By the Similarity Theorem, SNt = Nf (t). Let P = Mχf (K) and Q = SPS−1.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 34 / 41

SLIDE 74

Continuous nest algebras

Interpolation Theorem

Let N be the Volterra nest. For a Borel set S ⊆ [0, 1] write E(S) = MχS. Define the diagonal seminorm ix(T) := inf{P(Nx ⊖ Nt)TP(Nx ⊖ Nt) : t < x}

Theorem (Interpolation Theorem, [Orr95])

Let T ∈ Alg(N), a > 0, and S := {x : ix(T) ≥ a} Then there are A, B ∈ Alg(N) such that ATB = E(S).

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 35 / 41

SLIDE 75

Continuous nest algebras

Proof uses: Larson-Pitts [LP91] classification of idempotent equivalence Construction of “zero-diagonal” idempotents which sum to an idempotent that is equivalent to E(S) Factorization of “zero-diagonal” idempotents through T

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 36 / 41

SLIDE 76

Continuous nest algebras Ideals of continuous algebras

Corollary

Let N be a continuous nest and X ∈ Alg(N). TFAE:

1 There are A1 . . . , An and B1, . . . , Bn in Alg(N) such that

A1XB1 + · · · + ANXBn = I. I.e. X does not belong to any proper ideal of Alg(N).

2 There are A, B ∈ Alg(N) such that AXB = I. 3 it(X) ≥ a > 0 for all 0 ≤ t ≤ 1.

I.e. inf{P(Nt ⊖ Ns)TP(Nt ⊖ Ns) : 0 ≤ s < t ≤ I} > 0

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 37 / 41

SLIDE 77

Continuous nest algebras Ideals of continuous algebras

Corollary

Let N be a continuous nest and X ∈ Alg(N). TFAE:

1 There are A1 . . . , An and B1, . . . , Bn in Alg(N) such that

A1XB1 + · · · + ANXBn = I. I.e. X does not belong to any proper ideal of Alg(N).

2 There are A, B ∈ Alg(N) such that AXB = I. 3 it(X) ≥ a > 0 for all 0 ≤ t ≤ 1.

I.e. inf{P(Nt ⊖ Ns)TP(Nt ⊖ Ns) : 0 ≤ s < t ≤ I} > 0 Compare this with T where:

3. is analgous to X = I + S

We saw 1. ⇔ 2. We could not settle whether a version of 2. with two terms might be possible.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 37 / 41

SLIDE 78

Continuous nest algebras Ideals of continuous algebras

Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar]

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 38 / 41

SLIDE 79

Continuous nest algebras Ideals of continuous algebras

Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar]

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 38 / 41

SLIDE 80

Continuous nest algebras Ideals of continuous algebras

Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar]

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 38 / 41

SLIDE 81

Continuous nest algebras Ideals of continuous algebras

Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar]

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 38 / 41

SLIDE 82

Continuous nest algebras Ideals of continuous algebras

Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar]

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 38 / 41

SLIDE 83

Continuous nest algebras Ideals of continuous algebras

Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar]

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 38 / 41

SLIDE 84

Continuous nest algebras Epimorphisms

Davidson-Harrison-Orr, [DHO95] described “almost” all epimorphisms between nest algebras. Essentially one case was left open:

Question

Does there exist an epimorphism φ : T → B(H)?

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 39 / 41

SLIDE 85

Continuous nest algebras Epimorphisms

Davidson-Harrison-Orr, [DHO95] described “almost” all epimorphisms between nest algebras. Essentially one case was left open:

Question

Does there exist an epimorphism φ : T → B(H)?

Fact

If so, then ker φ contains an operator I + S (S ∈ S).

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 39 / 41

SLIDE 86

Continuous nest algebras Epimorphisms

Davidson-Harrison-Orr, [DHO95] described “almost” all epimorphisms between nest algebras. Essentially one case was left open:

Question

Does there exist an epimorphism φ : T → B(H)?

Fact

If so, then ker φ contains an operator I + S (S ∈ S).

Proof.

The commutator ideal of T is S and the commutator ideal of B(H) is B(H). Thus φ(S) = I = φ(I) and so I − S ∈ ker φ.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 39 / 41

SLIDE 87

Bibliography and other questions Stable rank

Definition

The Bass stable rank of an algebra is the smallest n such that whenever (g1, . . . , gn+1) generate the algebra as a left-ideal then we can find ai such that (g1 + b1gn+1, g2 + b2gn+1, . . . gn + bngn+1) also generate the algebra as a left ideal.

Question

What is the Bass stable rank of T ?

Theorem (Arveson, [Arv75])

G1, . . . , Gn generate T as a left ideal iff G ∗

1 P⊥ k G1 + · · · + G ∗ n P⊥ k Gn ≥ aPk

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 40 / 41

SLIDE 88

Bibliography and other questions Stable rank

http://www.math.unl.edu/∼jorr/presentations

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 41 / 41

SLIDE 89

Bibliography and other questions Bibliography

Joel Anderson. Extensions, restrictions, and representations of states on c∗-algebras. Transactions of the Amer. Math. Soc., 249(2):303–329, 1979. Joel Anderson. Extreme points in sets of positive linear maps on ⌊().

J. Func. Anal., 31(2):195–217, 1979.

William B. Arveson. Interpolation problems in nest algebras.

J. Func. Anal., 20:208–233, 1975.

Kenneth Berman, Herbert Halpern, Victor Kaftal, and Gary Weiss. Matrix norm inequalities and the relative dixmier property. Integral Equations Operator Theory, 1988.

J. Bourgain and L. Tzafriri.

On a problem of kadison and singer.

J. reine angew. Math., 1991.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 41 / 41

SLIDE 90

Bibliography and other questions Bibliography

Peter G. Casazza, Ole Christensen, Alexander M. Lindner, and Roman Vershynin. Frames and the feichtinger conjecture.

Proc. Amer. Math. Soc., 2005.

Peter G. Casazza and Janet C. Tremain. The paving conjecture is equivalent to the paving conjecture for triangular matrices. http://www.arxiv.org/abs/math.FA/0701101. Kenneth R. Davidson. Similarity and compact perturbations of nest algebras.

J. reine angew. Math., 348:286–294, 1984.

Kenneth R. Davidson. Nest Algebras, volume 191 of Res. Notes Math. Pitman, Boston, 1988. Kenneth R. Davidson, Kenneth J. Harrison, and John L. Orr. Epimorphisms of nest algebras.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 41 / 41

SLIDE 91

Bibliography and other questions Bibliography

Internat. J. Math., 1995.

Kenneth R. Davidson and John L. Orr. The invertibles are connected in infinite multiplicity nest algebras.

Bull. London Math. Soc., 1995.

Kenneth R. Davidson, John L. Orr, and David R. Pitts. Connectedness of the invertibles in certain nest algebras.

Canad. Math Bull., 1995.

John A. Erdos. Unitary invariants for nests. Pacific J. Math., 23:229–256, 1967. Richard V. Kadison and I. M. Singer. Extensions of pure states.

Amer. J. Math., 81:383–400, 1959.

David R. Larson. Nest algebras and similarity transformations.

Ann. Math., 121:409–427, 1985.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 41 / 41

SLIDE 92

Bibliography and other questions Bibliography

David R. Larson and David R. Pitts. Idempotents in nest algebras.

J. Func. Anal., 97:162–193, 1991.

John L. Orr. The maximal ideals of a nest algebra.

J. Func. Anal., 124:119–134, 1994.

John L. Orr. Triangular algebras and ideals of nest algebras. Memoirs of the Amer. Math. Soc., 562(117), 1995. John L. Orr. The stable ideals of a continuous nest algebra.

J. Operator Theory, 45:377–412, 2001.

John L. Orr. The stable ideals of a continuous nest algebra II.

J. Operator Theory, to appear.
G. A. Reid.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 41 / 41

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Bibliography and other questions Bibliography

On the calkin representations.

Proc. London Math. Soc., 23(3):547 – 564, 1971.

John R. Ringrose. On some algebras of operators.

Proc. London Math. Soc., 15(3):61–83, 1965.

John R. Ringrose. On some algebras of operators II.

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Nik Weaver. A counterexample to a conjecture of akemann and anderson.

Bull. London Math. Soc., 2003.

Nik Weaver. The kadison-singer problem in discrepancy theory. Discrete Math., 2004.

John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 41 / 41