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Enveloping semigroups of Rosenthal Banach spaces

Michael Megrelishvili (Bar-Ilan University) Co-author: Eli Glasner (Tel-Aviv University) Workshop on set theoretic methods in compact spaces and Banach spaces, Warsaw, April 21, 2013

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Most related references

1

• E. Glasner and M. Megrelishvili, Banach representations

and affine compactifications of dynamical systems. To appear in: the Fields institute proceedings dedicated to the 2010 thematic program on asymptotic geometric analysis.

2

• E. Glasner and M. Megrelishvili, Representations of

dynamical systems on Banach spaces not containing l1,

• Trans. AMS, 364 (2012), 6395-6424.

3

• E. Glasner, M. Megrelishvili and V.V. Uspenskij, On

metrizable enveloping semigroups, Israel J. of Math. 164 (2008), 317-332.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Motivation

There are many classical common points between DS and Banach spaces. Some new research lines:

• representations of DS on nice Banach spaces.
• Fragmentability concept and (non)sensitivity of DS.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Emphasis on

RECENT Applications:

• Introducing enveloping semigroups of Banach spaces

(inspired by de Leeuw-Glicksberg, Witz, Junghenn, ...)

• Representation of enveloping semigroups on Banach spaces.

(inspired by J. Pym, A. K¨

• hler, ...)
• Rosenthal compacta which are relevant for DS

(inspired by Bourgain-Fremlin-Talagrand (BFT) dichotomy)

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Emphasis on

RECENT Applications:

• Introducing enveloping semigroups of Banach spaces

(inspired by de Leeuw-Glicksberg, Witz, Junghenn, ...)

• Representation of enveloping semigroups on Banach spaces.

(inspired by J. Pym, A. K¨

• hler, ...)
• Rosenthal compacta which are relevant for DS

(inspired by Bourgain-Fremlin-Talagrand (BFT) dichotomy)

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

We will sketch Generalized Ellis thm: every compact right topological admissible group with fragmented left translations is a topological group

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Let S be a topologized semigroup (with e ∈ S). left translations λa : S → S, x → ax right transitions ρa : S → S, x → xa The subset Λ(S) := {a ∈ S : λa is continuous} is called the topological center of S. Definition A topologized semigroup S is said to be:

1

right topological semigroup if every ρa is continuous.

2

admissible if S is right topological and Λ(S) is dense in S.

3

semitopological if S × S → S is separately continuous.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Let S be a topologized semigroup (with e ∈ S). left translations λa : S → S, x → ax right transitions ρa : S → S, x → xa The subset Λ(S) := {a ∈ S : λa is continuous} is called the topological center of S. Definition A topologized semigroup S is said to be:

1

right topological semigroup if every ρa is continuous.

2

admissible if S is right topological and Λ(S) is dense in S.

3

semitopological if S × S → S is separately continuous.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Semigroup actions S × X → X, s · (t · x) = (st) · x e · x = x. Dynamical system (S, X) X is compact, S is semitopological and the action is (at least) separately continuous. Affine dynamical system (S, Q) convex Q ⊂ V ∈ LCS, ˜ s : X → X are affine s · (cu + (1 − c)v) = c(s · u) + (1 − c)(s · v) 0 ≤ c ≤ 1

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Semigroup actions S × X → X, s · (t · x) = (st) · x e · x = x. Dynamical system (S, X) X is compact, S is semitopological and the action is (at least) separately continuous. Affine dynamical system (S, Q) convex Q ⊂ V ∈ LCS, ˜ s : X → X are affine s · (cu + (1 − c)v) = c(s · u) + (1 − c)(s · v) 0 ≤ c ≤ 1

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Recall Ellis thm: every compact group with separately continuous multiplication map is a topological group

• Generalized Ellis thm: every compact right topological

admissible group with fragmented left translations is a topological group Corollary Let P be a compact admissible right topological group. Assume that P, as a topological space, is Fr´

• echet. Then P is a

topological group. In particular this holds if:

1

(Moors & Namioka) P is first countable.

2

(Namioka 72, Ruppert 73) P is metrizable.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Recall Ellis thm: every compact group with separately continuous multiplication map is a topological group

• Generalized Ellis thm: every compact right topological

admissible group with fragmented left translations is a topological group Corollary Let P be a compact admissible right topological group. Assume that P, as a topological space, is Fr´

• echet. Then P is a

topological group. In particular this holds if:

1

(Moors & Namioka) P is first countable.

2

(Namioka 72, Ruppert 73) P is metrizable.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Recall Ellis thm: every compact group with separately continuous multiplication map is a topological group

• Generalized Ellis thm: every compact right topological

admissible group with fragmented left translations is a topological group Corollary Let P be a compact admissible right topological group. Assume that P, as a topological space, is Fr´

• echet. Then P is a

topological group. In particular this holds if:

1

(Moors & Namioka) P is first countable.

2

(Namioka 72, Ruppert 73) P is metrizable.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Notation: V ∈ Ban = {Banach spaces}

• B∗ := BV ∗ (w∗-compact unit ball)
• Θ(V) := {σ ∈ L(V, V) : ||σ|| ≤ 1} (contr. operators)

Θ(V)s topological sem. wrt SOP Θ(V)w semitopological sem. wrt WOP

• Iso (V) ≤ Θ(V) linear onto isometries

Iso (V)s topological group. Iso (V)w semitopological group.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Notation: V ∈ Ban = {Banach spaces}

• B∗ := BV ∗ (w∗-compact unit ball)
• Θ(V) := {σ ∈ L(V, V) : ||σ|| ≤ 1} (contr. operators)

Θ(V)s topological sem. wrt SOP Θ(V)w semitopological sem. wrt WOP

• Iso (V) ≤ Θ(V) linear onto isometries

Iso (V)s topological group. Iso (V)w semitopological group.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

• Dual actions

For every S ≤ Θ(V)op (or, for every homomorphism h : S → Θ(V op)w) Lemma we have the dynamical system π : S × B∗ → B∗ (sep. cont. action) jointly continuous action if h is strongly continuous

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Vary V ∈ K ⊂ Ban K ∈ {Hilbert, Reflexive, Asplund, Rosenthal, ...} Suggests Hierarchies for:

• compact spaces
• actions on compact spaces
• top. (semi)groups
• functions

coming from functionals v ∈ V on DS X ⊂ V ∗ ˜ v : X → R, x → v, x

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Vary V ∈ K ⊂ Ban K ∈ {Hilbert, Reflexive, Asplund, Rosenthal, ...} Suggests Hierarchies for:

• compact spaces
• actions on compact spaces
• top. (semi)groups
• functions

coming from functionals v ∈ V on DS X ⊂ V ∗ ˜ v : X → R, x → v, x

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Topological prototype:

For (large) compact spaces X: Find a nice class K of Banach spaces such that there always is an element V ∈ K where X can be embedded into (V ∗, w∗) (or, into B∗ := BV ∗) compact sp. Banach sp. uEb = {uniformly Eberlein} Hilbert Eb = {Eberlein} Reflexive RN = {Radon-Nikodym} Asplund WRN = {weak Radon-Nikodym} Rosenthal (l1 ) Hilb ⊂ Refl ⊂ Aspl ⊂ Rosenthal ⊂ Ban uEb ⊂ Eb ⊂ RN ⊂ WRN ⊂ Comp

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Dynamical analog

Let f : X → X be a continuous function. Find: ”nice” V ∈ K, embedding α : X ֒ → B∗ ⊂ V ∗ and linear operator F ∈ L(V), ||F|| ≤ 1 such that f = F ∗|α(X) (F ∗ : V ∗ → V ∗ is the adjoint of F) X

α

• f

X

α

• B∗

F ∗ B∗

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Representations of DS on Banach spaces

• A representation of (S, X) on a Banach space V is a pair

h : S → Θ(V), α : X → V ∗ where h : S → Θ(V) is a co-homomorphism of semigroups and α : X → V ∗ is a weak∗ continuous (bounded) S-mapping with respect to the dual action S × V ∗ → V ∗ representation is weakly (strongly) continuous means that h is weakly (strongly) continuous. Faithful if α is a topological embedding.

• If S := G is a group then h : G → Iso (V)

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

• If K ⊂ Ban we say that a DS (S, X) is weakly (strongly)

K-representable if ∃ a weakly (resp., strongly) continuous faithful representation of (S, X) on a Banach space V ∈ K.

• An S-subspace of a direct product, of weakly (strongly)

K-representable S-spaces is said to be weakly (strongly) K-approximable.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Teleman’s representation

1

For every compact S-system X with the continuous action S × X → X we have the canonical faithful representation h : S → Θs, δ : X → V ∗

• n V := C(X)

x → δx δ(X) ⊂ P(X) ⊂ B∗. ⇓

2

For every topological group G ֒ → Iso (V)s with V := RUC(G) (and G → X is the Gelfand G-compactification induced by the algebra V := RUC(G)).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Geometric proof

Ellis Theorem Every compact semitopological group G is a topological group. Proof: Combine the following two theorems: Thm1 (Shtern, Me) If P is a semitopological semigroup then ∃ P ֒ → Θ(V) for some reflexive Banach sp. V. Thm2 WOP=SOP (Me) The weak and the strong operator topologies coincide on G ≤ Iso (V) for every Banach space V with PCP (e.g., for reflexive V, or the duals of Asplund spaces). By Thm1 G ≤ Iso (V)w ⊂ Θ(V)w for some reflexive V. By Thm 2 WOP=SOP on Iso (V) for every V ∈ PCP. So, G ≤ Iso (V)w = Iso (V)s is a topological group.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Geometric proof

Ellis Theorem Every compact semitopological group G is a topological group. Proof: Combine the following two theorems: Thm1 (Shtern, Me) If P is a semitopological semigroup then ∃ P ֒ → Θ(V) for some reflexive Banach sp. V. Thm2 WOP=SOP (Me) The weak and the strong operator topologies coincide on G ≤ Iso (V) for every Banach space V with PCP (e.g., for reflexive V, or the duals of Asplund spaces). By Thm1 G ≤ Iso (V)w ⊂ Θ(V)w for some reflexive V. By Thm 2 WOP=SOP on Iso (V) for every V ∈ PCP. So, G ≤ Iso (V)w = Iso (V)s is a topological group.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Geometric proof

Ellis Theorem Every compact semitopological group G is a topological group. Proof: Combine the following two theorems: Thm1 (Shtern, Me) If P is a semitopological semigroup then ∃ P ֒ → Θ(V) for some reflexive Banach sp. V. Thm2 WOP=SOP (Me) The weak and the strong operator topologies coincide on G ≤ Iso (V) for every Banach space V with PCP (e.g., for reflexive V, or the duals of Asplund spaces). By Thm1 G ≤ Iso (V)w ⊂ Θ(V)w for some reflexive V. By Thm 2 WOP=SOP on Iso (V) for every V ∈ PCP. So, G ≤ Iso (V)w = Iso (V)s is a topological group.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Let (S, X) be a DS with compact X defined by π : S × X → X f ∈ C(X) is said to be: AP (Almost Periodic) if cl(fS) is norm compact in C(X). WAP if clw(fS) is weakly compact in C(X). S-system X is said to be WAP if WAP(X)=C(X)

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Ellis-Lawson Joint Continuity Theorem: Let G be a subgroup of a compact semitopological monoid S. Suppose that S × X → X is a separately continuous action with compact X. Then G × X → X is jointly continuous (⇒ G is a topological group). Proof: (idea from [Me-Pestov-Uspenskij]) ∀f ∈ C(X) the orbit fS is p-compact and bounded hence (By Grothendieck’s Lemma) clw(fS) = fS is w-compact. So, C(X) = WAP(X). Hence (S, X) is a WAP system. Thm(Me 2003) Every WAP system admits sufficiently many reflexive representations. So, the proof of Ellis-Lawson’s thm can be reduced to the particular case where (S, X) = (Θ(V)op

w , BV ∗) for some reflexive

V with G := Iso (V). Now use again Iso (V)w = Iso (V)s, and Lemma 0.3 (for the last part take X := S and the natural action G × S → S).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Ellis-Lawson Joint Continuity Theorem: Let G be a subgroup of a compact semitopological monoid S. Suppose that S × X → X is a separately continuous action with compact X. Then G × X → X is jointly continuous (⇒ G is a topological group). Proof: (idea from [Me-Pestov-Uspenskij]) ∀f ∈ C(X) the orbit fS is p-compact and bounded hence (By Grothendieck’s Lemma) clw(fS) = fS is w-compact. So, C(X) = WAP(X). Hence (S, X) is a WAP system. Thm(Me 2003) Every WAP system admits sufficiently many reflexive representations. So, the proof of Ellis-Lawson’s thm can be reduced to the particular case where (S, X) = (Θ(V)op

w , BV ∗) for some reflexive

V with G := Iso (V). Now use again Iso (V)w = Iso (V)s, and Lemma 0.3 (for the last part take X := S and the natural action G × S → S).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Ellis-Lawson Joint Continuity Theorem: Let G be a subgroup of a compact semitopological monoid S. Suppose that S × X → X is a separately continuous action with compact X. Then G × X → X is jointly continuous (⇒ G is a topological group). Proof: (idea from [Me-Pestov-Uspenskij]) ∀f ∈ C(X) the orbit fS is p-compact and bounded hence (By Grothendieck’s Lemma) clw(fS) = fS is w-compact. So, C(X) = WAP(X). Hence (S, X) is a WAP system. Thm(Me 2003) Every WAP system admits sufficiently many reflexive representations. So, the proof of Ellis-Lawson’s thm can be reduced to the particular case where (S, X) = (Θ(V)op

w , BV ∗) for some reflexive

V with G := Iso (V). Now use again Iso (V)w = Iso (V)s, and Lemma 0.3 (for the last part take X := S and the natural action G × S → S).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Ellis-Lawson Joint Continuity Theorem: Let G be a subgroup of a compact semitopological monoid S. Suppose that S × X → X is a separately continuous action with compact X. Then G × X → X is jointly continuous (⇒ G is a topological group). Proof: (idea from [Me-Pestov-Uspenskij]) ∀f ∈ C(X) the orbit fS is p-compact and bounded hence (By Grothendieck’s Lemma) clw(fS) = fS is w-compact. So, C(X) = WAP(X). Hence (S, X) is a WAP system. Thm(Me 2003) Every WAP system admits sufficiently many reflexive representations. So, the proof of Ellis-Lawson’s thm can be reduced to the particular case where (S, X) = (Θ(V)op

w , BV ∗) for some reflexive

V with G := Iso (V). Now use again Iso (V)w = Iso (V)s, and Lemma 0.3 (for the last part take X := S and the natural action G × S → S).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Questions: what happens for ◮more general (than reflexive) Banach spaces ◮more general DS ◮general compact right topological (not necessarily semitopological) semigroups ? .... ◮What about representations of compact right topological semigroups (enveloping semigroups) on Banach spaces ?

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Questions: what happens for ◮more general (than reflexive) Banach spaces ◮more general DS ◮general compact right topological (not necessarily semitopological) semigroups ? .... ◮What about representations of compact right topological semigroups (enveloping semigroups) on Banach spaces ?

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Some classes of DS

• (Ellis-Nerurkar) (S, X) is WAP if C(X) = WAP(X).
• (K¨
• hler) (S, X) is tame (regular) if every f ∈ C(X) is regular,

i.e., fS does not contain an l1-sequence for every f ∈ C(X).

• (S, X) is HNS.

Here A ⊂ X is NS (Non-Sensitive) means that for every entourage ε from the unique compatible uniformity on X there exists an open subset O of X such that A ∩ O is nonempty and s(A ∩ O) is ε-small for every s ∈ S.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Some classes of DS

• (Ellis-Nerurkar) (S, X) is WAP if C(X) = WAP(X).
• (K¨
• hler) (S, X) is tame (regular) if every f ∈ C(X) is regular,

i.e., fS does not contain an l1-sequence for every f ∈ C(X).

• (S, X) is HNS.

Here A ⊂ X is NS (Non-Sensitive) means that for every entourage ε from the unique compatible uniformity on X there exists an open subset O of X such that A ∩ O is nonempty and s(A ∩ O) is ε-small for every s ∈ S.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Butterfly Effect

• Guckenheimer, Devaney, Auslander-Yorke, Glasner-Weiss, ...

(G, (X, d)) is sensitive if ∃ ε > 0 ∀ ∅ = O ⊂ X some gO is not ε-small

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Banach hierarchy for actions

Hilb ⊂ Refl ⊂ Aspl ⊂ Rosenthal ⊂ ...Ban uEb ⊂ Eb ⊂ RN ⊂ WRN ⊂ ...DS For compact metric dynamical S-systems X we have ”natural realizations” ◮[Me 2003] Eb = WAP ◮[Glasner-Me 2007] RN = HNS (Hereditarily Nonsensitive DS) ◮[Glasner-Me 2012] WRN = tame DS (for non-metrizable replace ”representable” by ”approximable”) * The crucial role of the Davis-Figiel-Johnson-Pelczy´ nski thm ...

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Banach hierarchy for actions

Hilb ⊂ Refl ⊂ Aspl ⊂ Rosenthal ⊂ ...Ban uEb ⊂ Eb ⊂ RN ⊂ WRN ⊂ ...DS For compact metric dynamical S-systems X we have ”natural realizations” ◮[Me 2003] Eb = WAP ◮[Glasner-Me 2007] RN = HNS (Hereditarily Nonsensitive DS) ◮[Glasner-Me 2012] WRN = tame DS (for non-metrizable replace ”representable” by ”approximable”) * The crucial role of the Davis-Figiel-Johnson-Pelczy´ nski thm ...

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Banach hierarchy for actions

Hilb ⊂ Refl ⊂ Aspl ⊂ Rosenthal ⊂ ...Ban uEb ⊂ Eb ⊂ RN ⊂ WRN ⊂ ...DS For compact metric dynamical S-systems X we have ”natural realizations” ◮[Me 2003] Eb = WAP ◮[Glasner-Me 2007] RN = HNS (Hereditarily Nonsensitive DS) ◮[Glasner-Me 2012] WRN = tame DS (for non-metrizable replace ”representable” by ”approximable”) * The crucial role of the Davis-Figiel-Johnson-Pelczy´ nski thm ...

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Definition E(X) := clp(j(S)) ⊂ X X the enveloping semigroup of (S, X) (where j(s) = λs : X → X. The associated homomorphism j : S → E(X) is a right topological semigroup compactification (the Ellis compactification) of S, j(e) = idX and the associated action πj : S × E(X) → E(X) is separately continuous. If the S-action

• n X is continuous then πj is continuous.

Remark Every enveloping semigroup E(S, X) is an example of a compact right topological admissible semigroup. Conversely, every compact right topological admissible semigroup P is an enveloping semigroup (of (Λ(P), P)).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Definition E(X) := clp(j(S)) ⊂ X X the enveloping semigroup of (S, X) (where j(s) = λs : X → X. The associated homomorphism j : S → E(X) is a right topological semigroup compactification (the Ellis compactification) of S, j(e) = idX and the associated action πj : S × E(X) → E(X) is separately continuous. If the S-action

• n X is continuous then πj is continuous.

Remark Every enveloping semigroup E(S, X) is an example of a compact right topological admissible semigroup. Conversely, every compact right topological admissible semigroup P is an enveloping semigroup (of (Λ(P), P)).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

• Bernoulli shift (Z, {0, 1}Z), σ(an) = (an−1).

E(Z, {0, 1}Z) = βZ.

• X := [0, 1] the unit interval, the action of the cyclic group Z on

X generated by the map f(x) = x2. Then E(Z, [0, 1]) is the 2-point compactification of Z.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

3 types of Separable Rosenthal compacta as E

• Ellis, following Furstenberg’s classical work, investigates the

projective action of GL(n, R) on the projective space Pn−1. It follows from his results that the corresponding enveloping semigroup is Frechet but not first countable.

• (a) Akin studies the action of G = GL(n, R) on the sphere

Sn−1 and shows that here the enveloping semigroup is first countable (but not hereditarily separable). (b) H+[0, 1] × [0, 1] → [0, 1]. E(H+, [0, 1]) ⊂ Helly compact first countable (but not hereditarily separable).

• A dynamical system (X, T) such that E(X, T) \ {T n}n∈Z is

homeomorphic to the two arrows space. Then E is a hereditarily separable Rosenthal compactum.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

3 types of Separable Rosenthal compacta as E

• Ellis, following Furstenberg’s classical work, investigates the

projective action of GL(n, R) on the projective space Pn−1. It follows from his results that the corresponding enveloping semigroup is Frechet but not first countable.

• (a) Akin studies the action of G = GL(n, R) on the sphere

Sn−1 and shows that here the enveloping semigroup is first countable (but not hereditarily separable). (b) H+[0, 1] × [0, 1] → [0, 1]. E(H+, [0, 1]) ⊂ Helly compact first countable (but not hereditarily separable).

• A dynamical system (X, T) such that E(X, T) \ {T n}n∈Z is

homeomorphic to the two arrows space. Then E is a hereditarily separable Rosenthal compactum.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

3 types of Separable Rosenthal compacta as E

• Ellis, following Furstenberg’s classical work, investigates the

projective action of GL(n, R) on the projective space Pn−1. It follows from his results that the corresponding enveloping semigroup is Frechet but not first countable.

• (a) Akin studies the action of G = GL(n, R) on the sphere

Sn−1 and shows that here the enveloping semigroup is first countable (but not hereditarily separable). (b) H+[0, 1] × [0, 1] → [0, 1]. E(H+, [0, 1]) ⊂ Helly compact first countable (but not hereditarily separable).

• A dynamical system (X, T) such that E(X, T) \ {T n}n∈Z is

homeomorphic to the two arrows space. Then E is a hereditarily separable Rosenthal compactum.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

3 types of Separable Rosenthal compacta as E

• Ellis, following Furstenberg’s classical work, investigates the

projective action of GL(n, R) on the projective space Pn−1. It follows from his results that the corresponding enveloping semigroup is Frechet but not first countable.

• (a) Akin studies the action of G = GL(n, R) on the sphere

Sn−1 and shows that here the enveloping semigroup is first countable (but not hereditarily separable). (b) H+[0, 1] × [0, 1] → [0, 1]. E(H+, [0, 1]) ⊂ Helly compact first countable (but not hereditarily separable).

• A dynamical system (X, T) such that E(X, T) \ {T n}n∈Z is

homeomorphic to the two arrows space. Then E is a hereditarily separable Rosenthal compactum.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

constructing many Rosenthal spaces

Remark These examples of DS are tame but not HNS. The Rosenthal representation of each of them gives a Rosenthal separable Banach space which is not Asplund. Once it was an important problem first resolved independently by James and Lindenstrauss-Stegall.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Questions for enveloping semigroups

{Semitopological compact semigroups} is a very restricted class of enveloping semigroups (=compact admissible right topological semigroups). Like WAP is a very restricted class of all DS. Question Let E(S, X) be the enveloping semigroup of a compact metrizable S-system X.

1

When does the compact space E(S, X) have nice topological properties (metrizable, Frechet-Urysohn) ?

2

When do the individual maps p : X → X, and λp : E(X) → E(X) for p ∈ E, have nice properties ?

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Question Which Polish groups G admit faithful right topological semigroup compactifications α : G → P such that P is: (a) semitopological; (b) metrizable; (c) first-countable ? (d) Frechet-Urysohn ?

• for G := H+[0, 1] any right topological semitopological or

metrizable semigroup compactifications are trivial. However, it admits a faithful right topological semigroup compactification H+[0, 1] ֒ → E(H+, [0, 1]) ⊂ Helly compact first countable Rosenthal compactum.

• By results of [Ferri-Galindo 2009] the group (c0, +) does not

admit faithful semitopological compactification.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Question Which Polish groups G admit faithful right topological semigroup compactifications α : G → P such that P is: (a) semitopological; (b) metrizable; (c) first-countable ? (d) Frechet-Urysohn ?

• for G := H+[0, 1] any right topological semitopological or

metrizable semigroup compactifications are trivial. However, it admits a faithful right topological semigroup compactification H+[0, 1] ֒ → E(H+, [0, 1]) ⊂ Helly compact first countable Rosenthal compactum.

• By results of [Ferri-Galindo 2009] the group (c0, +) does not

admit faithful semitopological compactification.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Question Which Polish groups G admit faithful right topological semigroup compactifications α : G → P such that P is: (a) semitopological; (b) metrizable; (c) first-countable ? (d) Frechet-Urysohn ?

• for G := H+[0, 1] any right topological semitopological or

metrizable semigroup compactifications are trivial. However, it admits a faithful right topological semigroup compactification H+[0, 1] ֒ → E(H+, [0, 1]) ⊂ Helly compact first countable Rosenthal compactum.

• By results of [Ferri-Galindo 2009] the group (c0, +) does not

admit faithful semitopological compactification.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Question Which Polish groups G admit faithful right topological semigroup compactifications α : G → P such that P is: (a) semitopological; (b) metrizable; (c) first-countable ? (d) Frechet-Urysohn ?

• for G := H+[0, 1] any right topological semitopological or

metrizable semigroup compactifications are trivial. However, it admits a faithful right topological semigroup compactification H+[0, 1] ֒ → E(H+, [0, 1]) ⊂ Helly compact first countable Rosenthal compactum.

• By results of [Ferri-Galindo 2009] the group (c0, +) does not

admit faithful semitopological compactification.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

• E is a group of cont. maps iff (G, X) is equicontinuous (AP).
• (Ellis) E is a group iff (G, X) is a distal system.
• (Ellis-Nerurkar) Every p : X → X for p ∈ E is a continuous

map iff (S, X) is a WAP system.

• Let (S, X) be point transitive. Then E is a semitopological

(topological) semigroup iff X is WAP (resp., AP).

• (Gl-Me-Uspenskij 2007) Let X be a compact metric S-system.

(a) E(S, X) is metrizable iff (S, X) is HNS. (b) Every p : X → X is Baire 1 iff (S, X) is tame.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

• E is a group of cont. maps iff (G, X) is equicontinuous (AP).
• (Ellis) E is a group iff (G, X) is a distal system.
• (Ellis-Nerurkar) Every p : X → X for p ∈ E is a continuous

map iff (S, X) is a WAP system.

• Let (S, X) be point transitive. Then E is a semitopological

(topological) semigroup iff X is WAP (resp., AP).

• (Gl-Me-Uspenskij 2007) Let X be a compact metric S-system.

(a) E(S, X) is metrizable iff (S, X) is HNS. (b) Every p : X → X is Baire 1 iff (S, X) is tame.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Representations of DS on Banach spaces

X compact metrizable dynamical S-system

DS classes Enveloping semigroup Banach representation WAP E(X) ⊂ C(X, X) Reflexive HNS E(X) is metrizable Asplund Tame ∀p ∈ E(X) is Baire 1 Rosenthal DS βN ⊂ E(X) is big Banach

♥ Special role of Tame Systems in the Dynamical version of Bourgain-Fremlin-Talagrand dichotomy.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Representations of DS on Banach spaces

X compact metrizable dynamical S-system

DS classes Enveloping semigroup Banach representation WAP E(X) ⊂ C(X, X) Reflexive HNS E(X) is metrizable Asplund Tame ∀p ∈ E(X) is Baire 1 Rosenthal DS βN ⊂ E(X) is big Banach

♥ Special role of Tame Systems in the Dynamical version of Bourgain-Fremlin-Talagrand dichotomy.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

A dynamical BFT dichotomy: Let X be a compact metric dynamical S-system. We have the following alternative. Either

1

E is a separable Rosenthal compact, hence card E ≤ 2ℵ0;

• r

2

the compact space E contains a homeomorphic copy of βN, hence card E = 22ℵ0. The first possibility holds iff X is a tame S-system.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

A dynamical BFT dichotomy: Let X be a compact metric dynamical S-system. We have the following alternative. Either

1

E is a separable Rosenthal compact, hence card E ≤ 2ℵ0;

• r

2

the compact space E contains a homeomorphic copy of βN, hence card E = 22ℵ0. The first possibility holds iff X is a tame S-system.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

• Bernoulli shift (Z, {0, 1}Z) is not Rosenthal representable (or,

approximable) Hint: The enveloping semigroup E = βZ.

• (Z, [0, 1]) with σ(t) = t2 is Asplund but not reflexively

representable. Hint: E is the 2-point compactification of Z (hence, metrizable).

• (H+[0, 1], [0, 1]) is Rosenthal but not Asplund representable.

Hint: Helly c. ⊃ E is a non-metrizable Rosenthal compactum.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Rosenthal compacta and enveloping semigroups

Answering a question of Talagrand, R. Pol gave an example of a separable compact Rosenthal space K which cannot be embedded in B1(X) (Baire 1 functions on X) for any compact metrizable X. Let’s say that a compact space K is

• strongly Rosenthal if K ⊂ B1(X) for some compact metr. X.
• admissible if there exists a compact metrizable space X and

a bounded subset F ⊂ C(X) with K ⊂ clp(F) ⊂ B1(X) where clp(F) is the pointwise closure of F in RX. admissible compactum ⇒ strongly Rosenthal When ⇐ ?

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Rosenthal compacta and enveloping semigroups

Answering a question of Talagrand, R. Pol gave an example of a separable compact Rosenthal space K which cannot be embedded in B1(X) (Baire 1 functions on X) for any compact metrizable X. Let’s say that a compact space K is

• strongly Rosenthal if K ⊂ B1(X) for some compact metr. X.
• admissible if there exists a compact metrizable space X and

a bounded subset F ⊂ C(X) with K ⊂ clp(F) ⊂ B1(X) where clp(F) is the pointwise closure of F in RX. admissible compactum ⇒ strongly Rosenthal When ⇐ ?

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Rosenthal compacta and enveloping semigroups

Answering a question of Talagrand, R. Pol gave an example of a separable compact Rosenthal space K which cannot be embedded in B1(X) (Baire 1 functions on X) for any compact metrizable X. Let’s say that a compact space K is

• strongly Rosenthal if K ⊂ B1(X) for some compact metr. X.
• admissible if there exists a compact metrizable space X and

a bounded subset F ⊂ C(X) with K ⊂ clp(F) ⊂ B1(X) where clp(F) is the pointwise closure of F in RX. admissible compactum ⇒ strongly Rosenthal When ⇐ ?

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem Let X be a compact metrizable S-system. Then (S, X) is tame iff the compactum K := E(X) is Rosenthal iff E(X) is a separable admissible compactum. Theorem (Rediscovering a result of Marciszewski) TFAE :

1

K is an admissible Rosenthal compactum.

2

K ֒ → V ∗∗ is homeomorphic to a weak∗ closed subset in the second dual of a separable Rosenthal Banach space V. One direction: B ⊂ clw∗(B) = B∗∗ ⊂ F(B∗) = B1(B∗) (Odell-Rosenthal).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem Let X be a compact metrizable S-system. Then (S, X) is tame iff the compactum K := E(X) is Rosenthal iff E(X) is a separable admissible compactum. Theorem (Rediscovering a result of Marciszewski) TFAE :

1

K is an admissible Rosenthal compactum.

2

K ֒ → V ∗∗ is homeomorphic to a weak∗ closed subset in the second dual of a separable Rosenthal Banach space V. One direction: B ⊂ clw∗(B) = B∗∗ ⊂ F(B∗) = B1(B∗) (Odell-Rosenthal).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Enveloping semigroup of a Banach space

• operator enveloping semigroup of a weakly cont.

homomorphism h : S → Θ(V)op E(h) := clw∗(h(S)) ⊂ clw∗(Θop) ⊂ L(V ∗) (Witz, Junghenn, Pym, Kohler) A net pi in L(V ∗) converges to p with respect to the weak∗

• perator topology iff pi(ψ) converges to p(ψ) in V ∗ with respect

to the weak∗ topology for each ψ ∈ V ∗. Or, equivalently, iff v, piψ → v, pψ ∀ v ∈ V, ψ ∈ V ∗.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Enveloping semigroup of a Banach space

• operator enveloping semigroup of a weakly cont.

homomorphism h : S → Θ(V)op E(h) := clw∗(h(S)) ⊂ clw∗(Θop) ⊂ L(V ∗) (Witz, Junghenn, Pym, Kohler) A net pi in L(V ∗) converges to p with respect to the weak∗

• perator topology iff pi(ψ) converges to p(ψ) in V ∗ with respect

to the weak∗ topology for each ψ ∈ V ∗. Or, equivalently, iff v, piψ → v, pψ ∀ v ∈ V, ψ ∈ V ∗.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Definition Define the enveloping semigroup of a Banach space V as E := clw∗(Θop) the weak∗ operator closure of Θ(V)op in L(V ∗). Alternatively, E can be defined as E(V) := E(Θ(V)op, B∗) Theorem V is reflexive iff E = Θop iff E(V) is a semitop. sem.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Lemma

1

E(V) is a compact right topological admissible affine semigroup and algebraically E(V) ≤ Θ(V ∗).

2

If V is separable then E(V) is separable. For every v ∈ SV and ψ ∈ SV ∗ we have:

3

Θv = vΘop = B.

4

vE = B∗∗.

5

clw∗(Θopψ) = B∗.

6

Eψ = B∗.

7

Λ(E) = Θop.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Fragmentability

Def: ([Namioka-Phelps], [Jayne-Rogers] (for f = id) [Jayne-Orihuela-Pallares-Vera] Let (X, τ) be a topological space and let (Y, µ) a uniform space.

• X is (τ, µ)-fragmented by a function f : X → Y if for every

nonempty subset A of X and every ε ∈ µ there exists an open subset O of X such that O ∩ A is nonempty and the set f(O ∩ A) is ε-small in Y. We also say in that case that the function f is fragmented. Notation: f ∈ F(X, Y).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Generalized versions for families of functions:

• We say that a family of functions F = {f : (X, τ) → (Y, µ)} is

fragmented (or, equi-fragmented) if the condition above holds simultaneously for all f ∈ F. That is, f(O ∩ A) is ε-small for every f ∈ F.

• We say that F is an eventually fragmented family if every

(countable) infinite subfamily F1 ⊂ F contains an infinite fragmented subfamily F2 ⊂ F1.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Fragmentability examples

1

(Namioka) Every weakly compact subset (X, τ) in a locally convex space V is (τ, ξ)-fragmented.

2

If X is Polish and Y is a separable metric space then f : X → Y is fragmented if and only if f is Baire 1.

3

When X is compact and (Y, ρ) metrizable uniform space then f : X → Y is fragmented iff f has a point of continuity property (i.e., for every closed nonempty A ⊂ X the restriction f|A : A → Y has a continuity point).

4

When Y is compact with its unique compatible uniformity µ then p : X → Y is fragmented if and only if f ◦ p : X → R has a point of continuity property for every f ∈ C(Y).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Example

1

Suppose F is a compact space, X is ˇ Cech-complete, Z is a uniform space and we are given a separately continuous map w : F × X → Z. Then the naturally associated family ˜ F := {˜ f : X → Z}f∈F is fragmented, where ˜ f(x) = w(f, x).

2

Suppose F is a compact and metrizable space, X is hereditarily Baire and M is separable and metrizable. Assume we are given a map w : F × X → M such that every ˜ x : F → M, f → w(f, x) is continuous and y : X → M is continuous at every ˜ y ∈ Y for some dense subset Y of

• F. Then the family ˜

F is fragmented. * B∗∗ × B∗ → R for separable Asplund V (take Y := B). * X compact metric G-space, E(X) × X → X (take Y := ˘ G := {˘ g : X → X}g∈G.)

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Example

1

Suppose F is a compact space, X is ˇ Cech-complete, Z is a uniform space and we are given a separately continuous map w : F × X → Z. Then the naturally associated family ˜ F := {˜ f : X → Z}f∈F is fragmented, where ˜ f(x) = w(f, x).

2

Suppose F is a compact and metrizable space, X is hereditarily Baire and M is separable and metrizable. Assume we are given a map w : F × X → M such that every ˜ x : F → M, f → w(f, x) is continuous and y : X → M is continuous at every ˜ y ∈ Y for some dense subset Y of

• F. Then the family ˜

F is fragmented. * B∗∗ × B∗ → R for separable Asplund V (take Y := B). * X compact metric G-space, E(X) × X → X (take Y := ˘ G := {˘ g : X → X}g∈G.)

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Example

1

Suppose F is a compact space, X is ˇ Cech-complete, Z is a uniform space and we are given a separately continuous map w : F × X → Z. Then the naturally associated family ˜ F := {˜ f : X → Z}f∈F is fragmented, where ˜ f(x) = w(f, x).

2

Suppose F is a compact and metrizable space, X is hereditarily Baire and M is separable and metrizable. Assume we are given a map w : F × X → M such that every ˜ x : F → M, f → w(f, x) is continuous and y : X → M is continuous at every ˜ y ∈ Y for some dense subset Y of

• F. Then the family ˜

F is fragmented. * B∗∗ × B∗ → R for separable Asplund V (take Y := B). * X compact metric G-space, E(X) × X → X (take Y := ˘ G := {˘ g : X → X}g∈G.)

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Example

1

Suppose F is a compact space, X is ˇ Cech-complete, Z is a uniform space and we are given a separately continuous map w : F × X → Z. Then the naturally associated family ˜ F := {˜ f : X → Z}f∈F is fragmented, where ˜ f(x) = w(f, x).

2

Suppose F is a compact and metrizable space, X is hereditarily Baire and M is separable and metrizable. Assume we are given a map w : F × X → M such that every ˜ x : F → M, f → w(f, x) is continuous and y : X → M is continuous at every ˜ y ∈ Y for some dense subset Y of

• F. Then the family ˜

F is fragmented. * B∗∗ × B∗ → R for separable Asplund V (take Y := B). * X compact metric G-space, E(X) × X → X (take Y := ˘ G := {˘ g : X → X}g∈G.)

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Asplund Spaces

Theorem TFAE:

1

V is an Asplund space.

2

V ∗ has the Radon-Nikod´ ym property.

3

Every bounded subset A of the dual V ∗ is (weak∗,norm)-fragmented.

4

B is a fragmented family of real valued maps on the compactum B∗. (4) is a reformulation of (3) in terms of fragmented families.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem TFAE:

1

V is a Rosenthal Banach space.

2

(E. Saab and P . Saab) B∗∗ ⊂ F(B∗).

3

(SS, Farmaki) B∗ is (weak∗, weak)-fragmented.

4

B is an eventually fragmented family of maps on B∗.

5

(Haydon) For every weak∗ compact subset Y ⊂ V ∗ the weak∗ and norm closures of the convex hull co(Y) in V ∗ coincide: clw∗(co(Y)) = clnorm(co(Y)). Condition (2) is a reformulation of a criterion from (SS). (4) uses results of Talagrand.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

A Banach space V is said to have the point of continuity property (PCP for short) if every bounded weakly closed subset C ⊂ V admits a point of continuity of the identity map (C, weak) → (C, norm) (Bourgain-Rosenthal, Edgar-Wheeler and Jayne-Rogers). Every Banach space with RNP has PCP . In particular, this is true for the duals of Asplund spaces and for reflexive spaces.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem (Jayne and Rogers) TFAE:

1

V has PCP .

2

Every bounded subset A ⊂ V is (weak, norm)-fragmented. Theorem [Me 2001] Let V be a Banach space with PCP (e.g., reflexive, RNP , or the dual of Asplund). Then for any bounded subgroup G of GL(V) (e.g., Iso (V)) the weak and strong operator topologies coincide on every orbit Gv of v ∈ V. Corollary (WOP=SOP) The weak and the strong operator topologies coincide on Iso (V) for every V ∈ PCP (e.g., for reflexive).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem (Jayne and Rogers) TFAE:

1

V has PCP .

2

Every bounded subset A ⊂ V is (weak, norm)-fragmented. Theorem [Me 2001] Let V be a Banach space with PCP (e.g., reflexive, RNP , or the dual of Asplund). Then for any bounded subgroup G of GL(V) (e.g., Iso (V)) the weak and strong operator topologies coincide on every orbit Gv of v ∈ V. Corollary (WOP=SOP) The weak and the strong operator topologies coincide on Iso (V) for every V ∈ PCP (e.g., for reflexive).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Tameness and HNS in terms of fragmentabity

(S, X) is tame iff ∀ p ∈ E(X) is a fragmented map (Baire 1, if X is metrizable) iff (S, X) admits sufficiently many representations

• n Rosenthal spaces.

(S, X) is HNS iff E(X) (equivalently, S) is equifragmented iff (S, X) admits sufficiently many representations on Asplund spaces.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Tameness and HNS in terms of fragmentabity

(S, X) is tame iff ∀ p ∈ E(X) is a fragmented map (Baire 1, if X is metrizable) iff (S, X) admits sufficiently many representations

• n Rosenthal spaces.

(S, X) is HNS iff E(X) (equivalently, S) is equifragmented iff (S, X) admits sufficiently many representations on Asplund spaces.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Smallness hierarchy (for orbits)

for functions f : X → R on a compact dynamical S-system X fS is weakly precompact f is WAP fS is fragmented f ∈ Asp(X) fS is eventually fragmented f ∈ Tame(X) F × X

ν

• α
• [−1, 1]

id

• VB × VB∗

[−1, 1]

For reflexive, Asplund, Rosenthal V, respectively.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Life beyond semitopological case

Definition A compact admissible right topological semigroup P is:

1

HNS-semigroup if {λa : P → P}a∈P is a fragmented family

• f maps.

2

tame semigroup if the left translation λa : P → P is a fragmented map for every a ∈ P. {compact semitopological semigroups} ⊂ {HNS-semigroups} ⊂ {Tame semigroups} ⊂ {c. admiss.}

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Lemma

1

Every compact semitopological semigroup P is a HNS-semigroup.

2

Every HNS-semigroup is tame.

3

If P is a metrizable compact right topological admissible semigroup then P is a HNS-semigroup.

4

If P is Fr´ echet (e.g., when it is Rosenthal), compact right topological admissible semigroup, then P is a tame semigroup.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem

1

E(V) is a semitopological semigroup iff V is reflexive.

2

E(V) is a HNS-semigroup iff V is Asplund.

3

E(V) is a tame semigroup iff V is Rosenthal.

4

If the compactum E(V) is Fr´ echet then V is a Rosenthal Banach space.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

1

E(V) is metrizable iff V is separable Asplund.

2

For separable V, E(V) is a Rosenthal compactum iff V is a Rosenthal Banach space.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Corollary (BFT dichotomy for separable Banach spaces) Let V be a separable Banach space and let E = E(V) be its (separable) enveloping semigroup. We have the following alternative. Either

1

E is a Rosenthal compactum, hence card E ≤ 2ℵ0; or

2

the compact space E contains a homeomorphic copy of βN, hence card E = 22ℵ0. The first possibility holds iff V is a Rosenthal Banach space.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem (Enveloping semigroup representation theorem)

1

Let P be a tame semigroup. Then there exists a Rosenthal Banach space V and a Λ(P)-admissible embedding of P into E(V).

2

If P is a HNS-semigroup then there is a Λ(P)-admissible embedding of P into E(V) where V is an Asplund Banach space.

3

If P is a semitopological semigroup then there is an embedding of P into Θ(V) = E(V ∗) where V is a reflexive Banach space. Important steps in the proof: Representations of DS (Λ(P), P) on Rosenthal spaces Haydon’s characterization of Rosenthal Banach spaces. E-compatibility.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem (Enveloping semigroup representation theorem)

1

Let P be a tame semigroup. Then there exists a Rosenthal Banach space V and a Λ(P)-admissible embedding of P into E(V).

2

If P is a HNS-semigroup then there is a Λ(P)-admissible embedding of P into E(V) where V is an Asplund Banach space.

3

If P is a semitopological semigroup then there is an embedding of P into Θ(V) = E(V ∗) where V is a reflexive Banach space. Important steps in the proof: Representations of DS (Λ(P), P) on Rosenthal spaces Haydon’s characterization of Rosenthal Banach spaces. E-compatibility.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

E-compatibility

Definition

1

By an affine S-compactification of an S-system X we mean a pair (α, Q), where α : X → Q is a continuous S-map and Q is a convex compact affine S-system such that α(X) affinely generates Q, that is co(α(X)) = Q.

2

We say that α : X → Q is E-compatible if the canonical continuous onto homomorphism Φ : E(Q) → E(Y) is an isomorphism, where Y := α(X).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

E-compatibility

Definition

1

By an affine S-compactification of an S-system X we mean a pair (α, Q), where α : X → Q is a continuous S-map and Q is a convex compact affine S-system such that α(X) affinely generates Q, that is co(α(X)) = Q.

2

We say that α : X → Q is E-compatible if the canonical continuous onto homomorphism Φ : E(Q) → E(Y) is an isomorphism, where Y := α(X).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Lemma Suppose cow∗(Y) = conorm(Y) holds for a weakly continuous representation (h, α) of a compact S-system X on a Banach space V, where Y := α(X). Then the affine compactification α : X → Q := co(Y) induced by the representation (h, α) is E-compatible. By Haydon’s thm we get that for Rosenthal V this holds always !

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Lemma Suppose cow∗(Y) = conorm(Y) holds for a weakly continuous representation (h, α) of a compact S-system X on a Banach space V, where Y := α(X). Then the affine compactification α : X → Q := co(Y) induced by the representation (h, α) is E-compatible. By Haydon’s thm we get that for Rosenthal V this holds always !

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Glasner’s theorem

Theorem Let α : X ֒ → Q be a faithful affine G-compactification of a compact minimal G-space X such that (G, Q) is distal. Then (G, Q) is equicontinuous (and hence, E(G, X) = E(G, Q) is a topological group). Originally, for metric X. Extended by D. Penazzi to non-metrizable case. Crucial step in the following generalization of Ellis thm.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Theorem (A generalized Ellis theorem) Every tame compact right topological group P is a topological group. Proof. By Represent. thm there exists a G-admissible embedding of P into E(V) for some Rosenthal Banach space V, where G := Λ(P) is a subgroup of P. P ֒ → Q := co(P) is E-compatible. P = E(G, P) = E(G, Q). So, E(G, Q) is also a group. Hence, (G, Q) is also distal. By Glasner’s thm we get (G, Q) is equicontinuous (observe that (G, P) is minimal). Then (G, P) is also equicontinuous and hence P = E(G, P) is a topological group.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Since every compact semitopological semigroup is tame, Ellis’ classical theorem (Fact 17) now follows as a special case of Theorem 0.25. (Note that we are not using Ellis’ theorem as an intermediate step in the proof of Theorem 0.25.) Corollary Let P be a compact admissible right topological group. Assume that P, as a topological space, is Fr´

• echet. Then P is a

topological group. Corollary (Glasner-2006 for metrizable X) A distal minimal (not necessarily, metric) compact G-system is tame if and only if it is equicontinuous.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Definition Compact admissible semigroup P is representable on a Banach space V if there exists an admissible embedding α of P into E(V) (admissibility means – ∃ a dense subsemigroup S ⊂ P s.t. α(S) ⊂ Θ(V)op = Λ(E)). Remark Compact admissible (semi)groups P not always admit admissible faithful representations on E(V) (In contrast to Topological groups and compact DS with continuous actions, Teleman’s representation 16) For example, in the case of Z → P := |D(Z)|, the maximal distal compactification of Z (P is a group) induced by the algebra of all distal functions on Z.

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Idea: (inspired by Todor˘ cevi´ c trichotomy in Rosenthal compacta) Investigate whether for separable Rosenthal Banach space V

• E(V) is hereditarily separable (non-metrizable, that is,

non-Asplund).

• E(V) is first countable (but not hereditarily separable)
• E(V) is not first countable.

Similar (and closely related) question for tame metric S-systems E(S, X).

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces

Thank you !

Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces