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 E. Glasner and M. Megrelishvili, Banach representations 1 and affine compactifications of dynamical systems. To appear in: the Fields institute proceedings dedicated to the 2010 thematic program on asymptotic geometric analysis. E. Glasner and M. Megrelishvili, Representations of 2 dynamical systems on Banach spaces not containing l 1 , Trans. AMS, 364 (2012), 6395-6424. E. Glasner, M. Megrelishvili and V.V. Uspenskij, On 3 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¨ ohler, ...) • 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¨ ohler, ...) • 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 λ a : S → S , x �→ ax right transitions ρ a : S → S , x �→ xa The subset Λ( S ) := { a ∈ S : λ a is continuous } is called the topological center of S . Definition A topologized semigroup S is said to be: right topological semigroup if every ρ a is continuous. 1 admissible if S is right topological and Λ( S ) is dense in S . 2 semitopological if S × S → S is separately continuous. 3 Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces
Let S be a topologized semigroup (with e ∈ S ). left translations λ a : S → S , x �→ ax right transitions ρ a : S → S , x �→ xa The subset Λ( S ) := { a ∈ S : λ a is continuous } is called the topological center of S . Definition A topologized semigroup S is said to be: right topological semigroup if every ρ a is continuous. 1 admissible if S is right topological and Λ( S ) is dense in S . 2 semitopological if S × S → S is separately continuous. 3 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: (Moors & Namioka) P is first countable. 1 (Namioka 72, Ruppert 73) P is metrizable. 2 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: (Moors & Namioka) P is first countable. 1 (Namioka 72, Ruppert 73) P is metrizable. 2 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: (Moors & Namioka) P is first countable. 1 (Namioka 72, Ruppert 73) P is metrizable. 2 Michael Megrelishvili Enveloping semigroups of Rosenthal Banach spaces
Notation: V ∈ Ban = { Banach spaces } • B ∗ := B V ∗ ( 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 ∗ := B V ∗ ( 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 ∗ := B V ∗ ) compact sp. Banach sp. uEb = { uniformly Eberlein } Hilbert Eb = { Eberlein } Reflexive RN = { Radon-Nikodym } Asplund WRN = { weak Radon-Nikodym } Rosenthal ( l 1 � ) 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 , → B ∗ ⊂ V ∗ embedding α : X ֒ and linear operator F ∈ L ( V ) , || F || ≤ 1 such that f = F ∗ | α ( X ) ( F ∗ : V ∗ → V ∗ is the adjoint of F ) f � X X α α F ∗ � B ∗ 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
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