Tame functions on topological groups and generalized (extreme) amenability

Transcription

1 Tame functions on topological groups and generalized (extreme) amenability Michael Megrelishvili (Bar-Ilan University) Ramat-Gan, Israel Joint project with Eli Glasner (Tel Aviv University) Banach Algebras and Applications Oulu, 2017

2 Main directions of this talk We study the algebra Tame(G) of tame functions on topological groups motivated by works of Pym, Köhler, Glasner, Rosenthal,... 1 Tame functions, algebra Tame(G) and dynamical systems 2 J. Pym s question and Operator enveloping semigroups 3 Generalized (extreme) amenability 4 Questions

3 Main directions of this talk We study the algebra Tame(G) of tame functions on topological groups motivated by works of Pym, Köhler, Glasner, Rosenthal,... 1 Tame functions, algebra Tame(G) and dynamical systems 2 J. Pym s question and Operator enveloping semigroups 3 Generalized (extreme) amenability 4 Questions

4 Main directions of this talk We study the algebra Tame(G) of tame functions on topological groups motivated by works of Pym, Köhler, Glasner, Rosenthal,... 1 Tame functions, algebra Tame(G) and dynamical systems 2 J. Pym s question and Operator enveloping semigroups 3 Generalized (extreme) amenability 4 Questions

5 Main directions of this talk We study the algebra Tame(G) of tame functions on topological groups motivated by works of Pym, Köhler, Glasner, Rosenthal,... 1 Tame functions, algebra Tame(G) and dynamical systems 2 J. Pym s question and Operator enveloping semigroups 3 Generalized (extreme) amenability 4 Questions

6 Main directions of this talk We study the algebra Tame(G) of tame functions on topological groups motivated by works of Pym, Köhler, Glasner, Rosenthal,... 1 Tame functions, algebra Tame(G) and dynamical systems 2 J. Pym s question and Operator enveloping semigroups 3 Generalized (extreme) amenability 4 Questions

7 Some other connections of Tame DS We do not touch 1 independence tuples [Kerr and Li 07] 2 NIP formulas in Logic [Shelah], [Ibarlucia], [Chernikov],... 3 (quasicrystals, tilings) [Aujogue15], [Aujogue-Kellendonk15],...

8 Notation Dynamical system (S, X ) X is compact, S is a semitopological semigroup and the action S X X, s (t x) = (st) x e x = x. is separately continuous. Reserve G := S for continuous group actions. j(s) = λ s : X X, j : S X X. Definition (Ellis compactification S E(X )) E(X ) := cl p (j(s)) X X the enveloping semigroup of (S, X ) j : S E(X ) right topological semigroup 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

9 Notation Dynamical system (S, X ) X is compact, S is a semitopological semigroup and the action S X X, s (t x) = (st) x e x = x. is separately continuous. Reserve G := S for continuous group actions. j(s) = λ s : X X, j : S X X. Definition (Ellis compactification S E(X )) E(X ) := cl p (j(s)) X X the enveloping semigroup of (S, X ) j : S E(X ) right topological semigroup 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

10 Classical definitions 1 G is extremely amenable if every continuous action on a compact space admits a fixed point. (Eq.: universal minimal G-system M(G) is trivial). Examples: U(l 2 ) (Gromov-Milman), H + [0, 1] (Pestov) Nonexamples: [Granirer-Lau] locally compact extremely amenable group is trivial. 2 G is amenable if every continuous affine action on a compact convex space admits a fixed point. (Eq.: universal irreducible affine G-system IA(G) is trivial). idea: let us free fixed point replacing it by a small G-system from some class P of compact G-systems.

11 Classical definitions 1 G is extremely amenable if every continuous action on a compact space admits a fixed point. (Eq.: universal minimal G-system M(G) is trivial). Examples: U(l 2 ) (Gromov-Milman), H + [0, 1] (Pestov) Nonexamples: [Granirer-Lau] locally compact extremely amenable group is trivial. 2 G is amenable if every continuous affine action on a compact convex space admits a fixed point. (Eq.: universal irreducible affine G-system IA(G) is trivial). idea: let us free fixed point replacing it by a small G-system from some class P of compact G-systems.

12 Classical definitions 1 G is extremely amenable if every continuous action on a compact space admits a fixed point. (Eq.: universal minimal G-system M(G) is trivial). Examples: U(l 2 ) (Gromov-Milman), H + [0, 1] (Pestov) Nonexamples: [Granirer-Lau] locally compact extremely amenable group is trivial. 2 G is amenable if every continuous affine action on a compact convex space admits a fixed point. (Eq.: universal irreducible affine G-system IA(G) is trivial). idea: let us free fixed point replacing it by a small G-system from some class P of compact G-systems.

13 Classical definitions 1 G is extremely amenable if every continuous action on a compact space admits a fixed point. (Eq.: universal minimal G-system M(G) is trivial). Examples: U(l 2 ) (Gromov-Milman), H + [0, 1] (Pestov) Nonexamples: [Granirer-Lau] locally compact extremely amenable group is trivial. 2 G is amenable if every continuous affine action on a compact convex space admits a fixed point. (Eq.: universal irreducible affine G-system IA(G) is trivial). idea: let us free fixed point replacing it by a small G-system from some class P of compact G-systems.

14 let us free fixed point (a little bit) Let P be a (nice) class of (small) compact G-spaces. We say that a topological group G is: 1 intrinsically P if every continuous action of G on a compact space X admits a compact G-subsystem Y X such that (G, Y ) P. 2 convexly intrinsically P if every continuous affine action of G on a compact affine space X admits a compact G-subsystem Y X such that (G, Y ) P. For P={one point trivial systems} If P {one point trivial systems} then (extreme) amenability extreme amenability intrinsically P amenability convexly intrinsically P

15 let us free fixed point (a little bit) Let P be a (nice) class of (small) compact G-spaces. We say that a topological group G is: 1 intrinsically P if every continuous action of G on a compact space X admits a compact G-subsystem Y X such that (G, Y ) P. 2 convexly intrinsically P if every continuous affine action of G on a compact affine space X admits a compact G-subsystem Y X such that (G, Y ) P. For P={one point trivial systems} If P {one point trivial systems} then (extreme) amenability extreme amenability intrinsically P amenability convexly intrinsically P

16 let us free fixed point (a little bit) Let P be a (nice) class of (small) compact G-spaces. We say that a topological group G is: 1 intrinsically P if every continuous action of G on a compact space X admits a compact G-subsystem Y X such that (G, Y ) P. 2 convexly intrinsically P if every continuous affine action of G on a compact affine space X admits a compact G-subsystem Y X such that (G, Y ) P. For P={one point trivial systems} If P {one point trivial systems} then (extreme) amenability extreme amenability intrinsically P amenability convexly intrinsically P

17 * we suggest in this context (and below try to justify) P = {Tame systems} Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Example G := Homeo + (T) is int-tame but not extremely amenable (even nonamenable).

18 * we suggest in this context (and below try to justify) P = {Tame systems} Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Example G := Homeo + (T) is int-tame but not extremely amenable (even nonamenable).

19 * we suggest in this context (and below try to justify) P = {Tame systems} Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Example G := Homeo + (T) is int-tame but not extremely amenable (even nonamenable).

20 * We explain why WAP and HNS do not work here. WAP= Weakly Almost Periodic DS HNS = Hereditarily NonSensitive DS

21 nothing new if P = WAP (or, even, if P =HNS) AP WAP HNS??? DS Lemma ( collapsing effect ) {convexly intrinsically AP} = {convexly intrinsically WAP} = {convexly intrinsically HNS} = usual amenability Proof. the algebra WAP(G) is amenable (Ryll-Nardzewski); the algebra Asp(G) (which determines the universal HNS G-compactification) is left amenable. Second explanation: Every minimal HNS (e.g., WAP) G-system is equicontinuous (AP), hence distal. Then it admits invariant probability measure (by Furstenberg s fixed point theorem, for instance).

22 nothing new if P = WAP (or, even, if P =HNS) AP WAP HNS??? DS Lemma ( collapsing effect ) {convexly intrinsically AP} = {convexly intrinsically WAP} = {convexly intrinsically HNS} = usual amenability Proof. the algebra WAP(G) is amenable (Ryll-Nardzewski); the algebra Asp(G) (which determines the universal HNS G-compactification) is left amenable. Second explanation: Every minimal HNS (e.g., WAP) G-system is equicontinuous (AP), hence distal. Then it admits invariant probability measure (by Furstenberg s fixed point theorem, for instance).

23 Notation: V Ban = {Banach spaces} B := B V (w -compact unit ball) Θ = Θ(V ) := {σ L(V, V ) : σ 1} (contr. operators) semitopological semigroup wrt WOP. Θ(V ) is compact iff V is reflexive. Iso(V ) Θ(V ) linear onto isometries Iso(V ) topological group wrt SOP Definition (enveloping semigroup of a Banach space V ) compact affine right topological semigroup E := cl w (Θ op ) (the weak operator closure of Θ(V ) op in L(V )). Alternatively, E := E(Θ(V ) op, B )

24 Notation: V Ban = {Banach spaces} B := B V (w -compact unit ball) Θ = Θ(V ) := {σ L(V, V ) : σ 1} (contr. operators) semitopological semigroup wrt WOP. Θ(V ) is compact iff V is reflexive. Iso(V ) Θ(V ) linear onto isometries Iso(V ) topological group wrt SOP Definition (enveloping semigroup of a Banach space V ) compact affine right topological semigroup E := cl w (Θ op ) (the weak operator closure of Θ(V ) op in L(V )). Alternatively, E := E(Θ(V ) op, B )

25 Definition (Dual actions) For every w-cont. (s-cont.) representation h : S Θ(V ) op we have sep. cont. (j. cont.) action π : S B B on (B, w ). Also we have operator compactification (Witz, Junghenn, [BJM]) S E h (S) = h(s) w E(V ). Question: Which actions S X X can be represented (embedded equivariantly into such dual actions) on nice Banach spaces V K Ban? S X X h α Θ op V V Question: Which c.r.t.s.c. S P can be represented as an operator compactification on...? α

26 Definition (Dual actions) For every w-cont. (s-cont.) representation h : S Θ(V ) op we have sep. cont. (j. cont.) action π : S B B on (B, w ). Also we have operator compactification (Witz, Junghenn, [BJM]) S E h (S) = h(s) w E(V ). Question: Which actions S X X can be represented (embedded equivariantly into such dual actions) on nice Banach spaces V K Ban? S X X h α Θ op V V Question: Which c.r.t.s.c. S P can be represented as an operator compactification on...? α

27 Definition (Dual actions) For every w-cont. (s-cont.) representation h : S Θ(V ) op we have sep. cont. (j. cont.) action π : S B B on (B, w ). Also we have operator compactification (Witz, Junghenn, [BJM]) S E h (S) = h(s) w E(V ). Question: Which actions S X X can be represented (embedded equivariantly into such dual actions) on nice Banach spaces V K Ban? S X X h α Θ op V V Question: Which c.r.t.s.c. S P can be represented as an operator compactification on...? α

28 Definition (Dual actions) For every w-cont. (s-cont.) representation h : S Θ(V ) op we have sep. cont. (j. cont.) action π : S B B on (B, w ). Also we have operator compactification (Witz, Junghenn, [BJM]) S E h (S) = h(s) w E(V ). Question: Which actions S X X can be represented (embedded equivariantly into such dual actions) on nice Banach spaces V K Ban? S X X h α Θ op V V Question: Which c.r.t.s.c. S P can be represented as an operator compactification on...? α

29 Some classes of Banach spaces Ref Asp Ros Ban Asp = Asplund Def: dual of every separable subspace is separable. I Reformulation: B is (w, norm)-fragmented ( nonempty A B and every ε > 0 weak-star open O V s.t. O A is nonempty and ε-small) II Reformulation: B is (equi)fragmented family of functions on B. Ros = Rosenthal Def: l 1 V I Reformulation: B C(B ) contains no independent subsequence. II Reformulation: B C(B ) is eventually (equi)fragmented.

30 Some classes of Banach spaces Ref Asp Ros Ban Asp = Asplund Def: dual of every separable subspace is separable. I Reformulation: B is (w, norm)-fragmented ( nonempty A B and every ε > 0 weak-star open O V s.t. O A is nonempty and ε-small) II Reformulation: B is (equi)fragmented family of functions on B. Ros = Rosenthal Def: l 1 V I Reformulation: B C(B ) contains no independent subsequence. II Reformulation: B C(B ) is eventually (equi)fragmented.

31 Some classes of Banach spaces Ref Asp Ros Ban Asp = Asplund Def: dual of every separable subspace is separable. I Reformulation: B is (w, norm)-fragmented ( nonempty A B and every ε > 0 weak-star open O V s.t. O A is nonempty and ε-small) II Reformulation: B is (equi)fragmented family of functions on B. Ros = Rosenthal Def: l 1 V I Reformulation: B C(B ) contains no independent subsequence. II Reformulation: B C(B ) is eventually (equi)fragmented.

32 Some classes of Banach spaces Ref Asp Ros Ban Asp = Asplund Def: dual of every separable subspace is separable. I Reformulation: B is (w, norm)-fragmented ( nonempty A B and every ε > 0 weak-star open O V s.t. O A is nonempty and ε-small) II Reformulation: B is (equi)fragmented family of functions on B. Ros = Rosenthal Def: l 1 V I Reformulation: B C(B ) contains no independent subsequence. II Reformulation: B C(B ) is eventually (equi)fragmented.

33 Some classes of Banach spaces Ref Asp Ros Ban Asp = Asplund Def: dual of every separable subspace is separable. I Reformulation: B is (w, norm)-fragmented ( nonempty A B and every ε > 0 weak-star open O V s.t. O A is nonempty and ε-small) II Reformulation: B is (equi)fragmented family of functions on B. Ros = Rosenthal Def: l 1 V I Reformulation: B C(B ) contains no independent subsequence. II Reformulation: B C(B ) is eventually (equi)fragmented.

34 Some classes of Banach spaces Ref Asp Ros Ban Asp = Asplund Def: dual of every separable subspace is separable. I Reformulation: B is (w, norm)-fragmented ( nonempty A B and every ε > 0 weak-star open O V s.t. O A is nonempty and ε-small) II Reformulation: B is (equi)fragmented family of functions on B. Ros = Rosenthal Def: l 1 V I Reformulation: B C(B ) contains no independent subsequence. II Reformulation: B C(B ) is eventually (equi)fragmented.

35 Families of functions and their representation Represent F X R on a Banach space with a good geometry F C(X ) bounded G-invariant. For example: F := fg...

36 Families of functions and their representation Represent F X R on a Banach space with a good geometry F C(X ) bounded G-invariant. For example: F := fg...

37 Small families of functions and their representation F X R V K ν α V V R id Theorem X be a compact S-system, F C(X ) bounded S-invariant. 1 (F, S, X ) admits a reflexive representation iff cl p (F ) C(X ) iff F has Grothendieck s DLP on X. 2 (F, S, X ) admits an Asplund representation iff F is a fragmented family. 3 (F, S, X ) admits a Rosenthal representation iff cl p (F ) F(X ) iff F contains no independent subset (iff F is an eventually fragmented family).

38 Small families of functions and their representation F X R V K ν α V V R id Theorem X be a compact S-system, F C(X ) bounded S-invariant. 1 (F, S, X ) admits a reflexive representation iff cl p (F ) C(X ) iff F has Grothendieck s DLP on X. 2 (F, S, X ) admits an Asplund representation iff F is a fragmented family. 3 (F, S, X ) admits a Rosenthal representation iff cl p (F ) F(X ) iff F contains no independent subset (iff F is an eventually fragmented family).

39 Small families of functions and their representation F X R V K ν α V V R id Theorem X be a compact S-system, F C(X ) bounded S-invariant. 1 (F, S, X ) admits a reflexive representation iff cl p (F ) C(X ) iff F has Grothendieck s DLP on X. 2 (F, S, X ) admits an Asplund representation iff F is a fragmented family. 3 (F, S, X ) admits a Rosenthal representation iff cl p (F ) F(X ) iff F contains no independent subset (iff F is an eventually fragmented family).

40 Small families of functions and their representation F X R V K ν α V V R id Theorem X be a compact S-system, F C(X ) bounded S-invariant. 1 (F, S, X ) admits a reflexive representation iff cl p (F ) C(X ) iff F has Grothendieck s DLP on X. 2 (F, S, X ) admits an Asplund representation iff F is a fragmented family. 3 (F, S, X ) admits a Rosenthal representation iff cl p (F ) F(X ) iff F contains no independent subset (iff F is an eventually fragmented family).

41 Representations of DS on Banach spaces representations of DS S X X on Hilb Ref Asp Ros Ban ueb Eb RN WRN Comp Banach spaces Compact spaces Hilb r WAP HNS Tame DS Dynamical systems WAP = Reflexively-approximable DS Metrizable-WAP = reflexively-representable DS......

42 Representations of DS on Banach spaces representations of DS S X X on Hilb Ref Asp Ros Ban ueb Eb RN WRN Comp Banach spaces Compact spaces Hilb r WAP HNS Tame DS Dynamical systems WAP = Reflexively-approximable DS Metrizable-WAP = reflexively-representable DS......

43 Representations of DS on Banach spaces representations of DS S X X on Hilb Ref Asp Ros Ban ueb Eb RN WRN Comp Banach spaces Compact spaces Hilb r WAP HNS Tame DS Dynamical systems WAP = Reflexively-approximable DS Metrizable-WAP = reflexively-representable DS......

44 Representations of DS on Banach spaces representations of DS S X X on Hilb Ref Asp Ros Ban ueb Eb RN WRN Comp Banach spaces Compact spaces Hilb r WAP HNS Tame DS Dynamical systems WAP = Reflexively-approximable DS Metrizable-WAP = reflexively-representable DS......

45 Matrix coefficients For s-cont. co-representations h : G Iso(V ), V K we have matrix coefficient algebras WAP(G) Asp(G) Tame(G) RUC(G) WAP(G) = Mat G (Reflexive) Asp(G) = Mat G (Asplund) Tame(G) = Mat G (Rosenthal)... RUC(G) = Mat G (Banach)

46 Matrix coefficients For s-cont. co-representations h : G Iso(V ), V K we have matrix coefficient algebras WAP(G) Asp(G) Tame(G) RUC(G) WAP(G) = Mat G (Reflexive) Asp(G) = Mat G (Asplund) Tame(G) = Mat G (Rosenthal)... RUC(G) = Mat G (Banach)

47 The Origin of tameness: Questions of Pym and Köhler Let X be a compact S-space and P(X ) C(X ) be the space of all probability measures on X. We have usual X P(X ) = co(x ) w. 1 J.S. Pym 90: The relationship between q : E(P(X )) E(X ) is not clear... (when q is isomorphic? eq.: injective) 2 A. Köhler 95: E(P(X )) = E(S) weak-star operator closure of the adjoint semigroup S op (enveloping operator semigroup). Defined tameness and proved that it is a sufficient condition. 3 S. Immervoll 99: counterexample (for some special S). 4 E. Glasner 06-07: counterexample for cascades S := Z and direct proof of Köhler s thm.

48 The Origin of tameness: Questions of Pym and Köhler Let X be a compact S-space and P(X ) C(X ) be the space of all probability measures on X. We have usual X P(X ) = co(x ) w. 1 J.S. Pym 90: The relationship between q : E(P(X )) E(X ) is not clear... (when q is isomorphic? eq.: injective) 2 A. Köhler 95: E(P(X )) = E(S) weak-star operator closure of the adjoint semigroup S op (enveloping operator semigroup). Defined tameness and proved that it is a sufficient condition. 3 S. Immervoll 99: counterexample (for some special S). 4 E. Glasner 06-07: counterexample for cascades S := Z and direct proof of Köhler s thm.

49 The Origin of tameness: Questions of Pym and Köhler Let X be a compact S-space and P(X ) C(X ) be the space of all probability measures on X. We have usual X P(X ) = co(x ) w. 1 J.S. Pym 90: The relationship between q : E(P(X )) E(X ) is not clear... (when q is isomorphic? eq.: injective) 2 A. Köhler 95: E(P(X )) = E(S) weak-star operator closure of the adjoint semigroup S op (enveloping operator semigroup). Defined tameness and proved that it is a sufficient condition. 3 S. Immervoll 99: counterexample (for some special S). 4 E. Glasner 06-07: counterexample for cascades S := Z and direct proof of Köhler s thm.

50 The Origin of tameness: Questions of Pym and Köhler Let X be a compact S-space and P(X ) C(X ) be the space of all probability measures on X. We have usual X P(X ) = co(x ) w. 1 J.S. Pym 90: The relationship between q : E(P(X )) E(X ) is not clear... (when q is isomorphic? eq.: injective) 2 A. Köhler 95: E(P(X )) = E(S) weak-star operator closure of the adjoint semigroup S op (enveloping operator semigroup). Defined tameness and proved that it is a sufficient condition. 3 S. Immervoll 99: counterexample (for some special S). 4 E. Glasner 06-07: counterexample for cascades S := Z and direct proof of Köhler s thm.

51 The Origin of tameness: Questions of Pym and Köhler Let X be a compact S-space and P(X ) C(X ) be the space of all probability measures on X. We have usual X P(X ) = co(x ) w. 1 J.S. Pym 90: The relationship between q : E(P(X )) E(X ) is not clear... (when q is isomorphic? eq.: injective) 2 A. Köhler 95: E(P(X )) = E(S) weak-star operator closure of the adjoint semigroup S op (enveloping operator semigroup). Defined tameness and proved that it is a sufficient condition. 3 S. Immervoll 99: counterexample (for some special S). 4 E. Glasner 06-07: counterexample for cascades S := Z and direct proof of Köhler s thm.

52 Independent sequences of functions Definition {f n : X R} n N is independent if a < b s.t. n P fn 1 (, a) n M for all finite disjoint subsets P, M of N. fn 1 (b, ) Example The sequence of projections on the Cantor set {π m : {0, 1} Z {0, 1}} m Z is independent. Pointwise closure of this family is βz (remember Bourgain-Fremlin-Talagrand dichotomy).

53 Independent sequences of functions Definition {f n : X R} n N is independent if a < b s.t. n P fn 1 (, a) n M for all finite disjoint subsets P, M of N. fn 1 (b, ) Example The sequence of projections on the Cantor set {π m : {0, 1} Z {0, 1}} m Z is independent. Pointwise closure of this family is βz (remember Bourgain-Fremlin-Talagrand dichotomy).

54 Tame systems Köhler s definition Definition (Köhler95) f C(X ) is said to be regular (tame, in the terminology of Glasner) if the family of functions fg does not contain an independent sequence. Notation: f Tame(X ). (G, X ) is tame if Tame(X ) = C(X ). Example why (Z, {0, 1} Z ) is not tame: the projection π 0 : {0, 1} Z {0, 1} is not a tame function (because π 0 G = {π k : k Z} is independent).

55 Definition f Tame(G) if f RUC(G) is tame and fg does not contain an independent sequence of functions. * Equivalently: The Gelfand G-space X f induced by the Banach G-subalgebra A f :=< fg > of RUC(G) is a tame DS. ** If G is second countable it is equivalent to say that carde(x f ) 2 ℵ 0 (So, Tame(G) = non-monster type functions).

56 Definition f Tame(G) if f RUC(G) is tame and fg does not contain an independent sequence of functions. * Equivalently: The Gelfand G-space X f induced by the Banach G-subalgebra A f :=< fg > of RUC(G) is a tame DS. ** If G is second countable it is equivalent to say that carde(x f ) 2 ℵ 0 (So, Tame(G) = non-monster type functions).

57 Theorem (GM13 Representation thm for affine compactifications) A metric G-system X is tame if and only if (G, X ) admits a representation on a Rosenthal space V ( equiv. pair h : G Iso(V ), α : X V ). Furthermore, one may assume that X P(X ) = co(x ) w is equivalent to α(x ) co(α(x )) w * If X is not necessarily metrizable then representability should be replaced by approximability.

58 Theorem (GM13 Representation thm for affine compactifications) A metric G-system X is tame if and only if (G, X ) admits a representation on a Rosenthal space V ( equiv. pair h : G Iso(V ), α : X V ). Furthermore, one may assume that X P(X ) = co(x ) w is equivalent to α(x ) co(α(x )) w * If X is not necessarily metrizable then representability should be replaced by approximability.

59 Theorem (GM13 Representation thm for affine compactifications) A metric G-system X is tame if and only if (G, X ) admits a representation on a Rosenthal space V ( equiv. pair h : G Iso(V ), α : X V ). Furthermore, one may assume that X P(X ) = co(x ) w is equivalent to α(x ) co(α(x )) w * If X is not necessarily metrizable then representability should be replaced by approximability.

60 geometric proof of Köhler s Thm Theorem (Haydon76) Banach space V is Rosenthal (l 1 V ) iff co w (Y ) = co norm (Y ) for every weak-star compact Y V. Combining with representation thm we can derive geometric proof of Köhler s Thm: every tame (not necessarily metrizable) system is injective satisfies the condition of Pym.

61 geometric proof of Köhler s Thm Theorem (Haydon76) Banach space V is Rosenthal (l 1 V ) iff co w (Y ) = co norm (Y ) for every weak-star compact Y V. Combining with representation thm we can derive geometric proof of Köhler s Thm: every tame (not necessarily metrizable) system is injective satisfies the condition of Pym.

62 E(X ) in general Let X be a compact metrizable G-system. in general: 1 it is possible that card(e(x )) = 2 2ℵ 0 (always card(e(x )) 2 2ℵ 0 for metrizable X ) 2 The topology of E described by nets: g i p E(X ) g i x px x X (not always by sequences!) 3 p : X X is not Baire 1 class function. Example (Bernoulli shift Z-system) Z {0, 1} Z {0, 1} Z, (n, (x k )) := (x k+n ). Then E(X ) = βz.

63 E(X ) in general Let X be a compact metrizable G-system. in general: 1 it is possible that card(e(x )) = 2 2ℵ 0 (always card(e(x )) 2 2ℵ 0 for metrizable X ) 2 The topology of E described by nets: g i p E(X ) g i x px x X (not always by sequences!) 3 p : X X is not Baire 1 class function. Example (Bernoulli shift Z-system) Z {0, 1} Z {0, 1} Z, (n, (x k )) := (x k+n ). Then E(X ) = βz.

64 Tame = largest class of all small systems Theorem (Dynamical BFT-dichotomy - GM06 ) For a compact metric S-system X we have the following dichotomy. Either 1 E(X ) satisfies the Heine principle (cl(a) = scl(a)) (and hence card(e) 2 ℵ 0 ); or 2 E(X ) contains a topological copy of βz (and hence card(e) = 2 2ℵ 0 as big as possible );.

65 Tame = largest class of all small systems Theorem (Dynamical BFT-dichotomy - GM06 ) For a compact metric S-system X we have the following dichotomy. Either 1 E(X ) satisfies the Heine principle (cl(a) = scl(a)) (and hence card(e) 2 ℵ 0 ); or 2 E(X ) contains a topological copy of βz (and hence card(e) = 2 2ℵ 0 as big as possible );.

66 (Ellis-Nerurkar) (S, X ) is WAP iff E C(X ). (In this case E is a semitopological semigroup) Lemma (TFAE:) 1 (S, X ) is tame. 2 every left translation λ a : E(X ) E(X ) is a fragmented map (eq.: f a : X R has PCP for every f C(E(X ))). 3 j : S E(X ) is equivalent to a Rosenthal operator compactification. Example V is Rosenthal iff E(V ) is a tame semigroup (cond. 2 of Lemma)

67 (Ellis-Nerurkar) (S, X ) is WAP iff E C(X ). (In this case E is a semitopological semigroup) Lemma (TFAE:) 1 (S, X ) is tame. 2 every left translation λ a : E(X ) E(X ) is a fragmented map (eq.: f a : X R has PCP for every f C(E(X ))). 3 j : S E(X ) is equivalent to a Rosenthal operator compactification. Example V is Rosenthal iff E(V ) is a tame semigroup (cond. 2 of Lemma)

68 (Ellis-Nerurkar) (S, X ) is WAP iff E C(X ). (In this case E is a semitopological semigroup) Lemma (TFAE:) 1 (S, X ) is tame. 2 every left translation λ a : E(X ) E(X ) is a fragmented map (eq.: f a : X R has PCP for every f C(E(X ))). 3 j : S E(X ) is equivalent to a Rosenthal operator compactification. Example V is Rosenthal iff E(V ) is a tame semigroup (cond. 2 of Lemma)

69 In contrast Example c.r.s.c. Z Z D(Z) is not an operator compactification, where D(Z) is the algebra of all distal functions.

70 More examples Theorem Circularly ordered DS are tame (in fact, Rosenthal representable). For example: (H + (T), T), Sturmian like symbolic systems. G := H + [0, 1] is reflexively (even, Asplund) trivial. Any right topological semitopological or metrizable semigroup compactifications of such G are trivial. However, it is Rosenthal representable and admits a top. faithful right topological semigroup compactification H + [0, 1] E(H +, [0, 1]) Helly compact which is first countable. By results of [Ferri-Galindo 09] the group (c 0, +) does not admit top. faithful reflexive repr. (or, semitop. compactification)

71 More examples Theorem Circularly ordered DS are tame (in fact, Rosenthal representable). For example: (H + (T), T), Sturmian like symbolic systems. G := H + [0, 1] is reflexively (even, Asplund) trivial. Any right topological semitopological or metrizable semigroup compactifications of such G are trivial. However, it is Rosenthal representable and admits a top. faithful right topological semigroup compactification H + [0, 1] E(H +, [0, 1]) Helly compact which is first countable. By results of [Ferri-Galindo 09] the group (c 0, +) does not admit top. faithful reflexive repr. (or, semitop. compactification)

72 More examples Theorem Circularly ordered DS are tame (in fact, Rosenthal representable). For example: (H + (T), T), Sturmian like symbolic systems. G := H + [0, 1] is reflexively (even, Asplund) trivial. Any right topological semitopological or metrizable semigroup compactifications of such G are trivial. However, it is Rosenthal representable and admits a top. faithful right topological semigroup compactification H + [0, 1] E(H +, [0, 1]) Helly compact which is first countable. By results of [Ferri-Galindo 09] the group (c 0, +) does not admit top. faithful reflexive repr. (or, semitop. compactification)

73 More examples Theorem Circularly ordered DS are tame (in fact, Rosenthal representable). For example: (H + (T), T), Sturmian like symbolic systems. G := H + [0, 1] is reflexively (even, Asplund) trivial. Any right topological semitopological or metrizable semigroup compactifications of such G are trivial. However, it is Rosenthal representable and admits a top. faithful right topological semigroup compactification H + [0, 1] E(H +, [0, 1]) Helly compact which is first countable. By results of [Ferri-Galindo 09] the group (c 0, +) does not admit top. faithful reflexive repr. (or, semitop. compactification)

74 Tame Symbolic systems Example Sturmian like symbolic systems are c-ordered, hence, tame (not HNS). Sturmian coding c : Z {0, 1} (e.g., Fibonacci sequence defined by Fibonacci substitution: s 0 = 0, s 1 = 01, s n = s n 1 s n 2 ) (being tame) can be represented on a Rosenthal Banach space. There exist: a Rosenthal Banach space V, a linear isometry σ Iso(V ) and two vectors v V, ϕ V c n = σ n (v), ϕ = ϕ(σ n (v)) n Z. * cannot be represented on Asplund (Reflexive) spaces!

75 Tame Symbolic systems Example Sturmian like symbolic systems are c-ordered, hence, tame (not HNS). Sturmian coding c : Z {0, 1} (e.g., Fibonacci sequence defined by Fibonacci substitution: s 0 = 0, s 1 = 01, s n = s n 1 s n 2 ) (being tame) can be represented on a Rosenthal Banach space. There exist: a Rosenthal Banach space V, a linear isometry σ Iso(V ) and two vectors v V, ϕ V c n = σ n (v), ϕ = ϕ(σ n (v)) n Z. * cannot be represented on Asplund (Reflexive) spaces!

76 Tame Symbolic systems Example Sturmian like symbolic systems are c-ordered, hence, tame (not HNS). Sturmian coding c : Z {0, 1} (e.g., Fibonacci sequence defined by Fibonacci substitution: s 0 = 0, s 1 = 01, s n = s n 1 s n 2 ) (being tame) can be represented on a Rosenthal Banach space. There exist: a Rosenthal Banach space V, a linear isometry σ Iso(V ) and two vectors v V, ϕ V c n = σ n (v), ϕ = ϕ(σ n (v)) n Z. * cannot be represented on Asplund (Reflexive) spaces!

77 Tame Symbolic systems Example Sturmian like symbolic systems are c-ordered, hence, tame (not HNS). Sturmian coding c : Z {0, 1} (e.g., Fibonacci sequence defined by Fibonacci substitution: s 0 = 0, s 1 = 01, s n = s n 1 s n 2 ) (being tame) can be represented on a Rosenthal Banach space. There exist: a Rosenthal Banach space V, a linear isometry σ Iso(V ) and two vectors v V, ϕ V c n = σ n (v), ϕ = ϕ(σ n (v)) n Z. * cannot be represented on Asplund (Reflexive) spaces!

78 Enveloping semigroup of Sturmian systems Example Sturmian system X α is a circularly ordered Z-system embedded into the c-ordered lexicographic order T T := T {, +} (split any point of the orbit of 0 on T ). Moreover, the enveloping (tame) semigroup E(X α ) is a circularly ordered cascade T T Z (not metrizable). It contains T T as its unique minimal Z-subspace. Every point of Z in E is isolated. E = T T {σ n : n Z}, where (T T, σ) is Ellis double circle cascade: T T = {β ± : β T = [0, 1)} and σ β ± = (β + α) ±. β + 1 β± 2 = (β 1 + β 2 ) + β 1 β± 2 = (β 1 + β 2 ) E = T T Z T {, 0, +} c-ordered lexicographic order [nα, σ n, nα + ] the interval (nα, nα + ) E contains only the single element σ n (so, isolated)

79 Figure: c-ordered lexicographic product (from Wikipedia)

80 from fixed point to a tame subsystem We propose P=Tame We say that a topological group G is: 1 intrinsically tame if every continuous action of G on a compact space X admits a compact G-subsystem Y X which is tame. 2 convexly intrinsically tame if every continuous affine action of G on a compact affine space X admits a compact G-subsystem Y X which is tame. extreme amenability int-tame amenability conv-int-tame

81 from fixed point to a tame subsystem We propose P=Tame We say that a topological group G is: 1 intrinsically tame if every continuous action of G on a compact space X admits a compact G-subsystem Y X which is tame. 2 convexly intrinsically tame if every continuous affine action of G on a compact affine space X admits a compact G-subsystem Y X which is tame. extreme amenability int-tame amenability conv-int-tame

82 from fixed point to a tame subsystem We propose P=Tame We say that a topological group G is: 1 intrinsically tame if every continuous action of G on a compact space X admits a compact G-subsystem Y X which is tame. 2 convexly intrinsically tame if every continuous affine action of G on a compact affine space X admits a compact G-subsystem Y X which is tame. extreme amenability int-tame amenability conv-int-tame

83 Example G := Homeo + (T) is int-tame but not extremely amenable (even nonamenable). Sketch: M(G) = T (by V. Pestov 98). Circular-order preserving dynamical systems are tame. For G := Homeo + (T) in any compact G-space we can find a G-circle or a fixed point (which are tame)

84 Example G := Homeo + (T) is int-tame but not extremely amenable (even nonamenable). Sketch: M(G) = T (by V. Pestov 98). Circular-order preserving dynamical systems are tame. For G := Homeo + (T) in any compact G-space we can find a G-circle or a fixed point (which are tame)

85 Example G := Homeo + (T) is int-tame but not extremely amenable (even nonamenable). Sketch: M(G) = T (by V. Pestov 98). Circular-order preserving dynamical systems are tame. For G := Homeo + (T) in any compact G-space we can find a G-circle or a fixed point (which are tame)

86 Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Sketch: for n = 2. IA(G) = P(K) probablity measures on K, where K is the 1-dimensional real projective space n = 2 In any compact affine G-space we can observe: 1-dim real projective G-space or a fixed point (which are tame). for n 2: flag manifolds and their G-quotients. Remark (Veech, Chou G = SL n (R) is Minimally WAP ) Universal semitopological G-compactification G WAP = G { } = 1-point compactification. WAP(SL n (R)) = c 0 (G) R. Remark (In contrast) G Tame is not metrizable.

87 Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Sketch: for n = 2. IA(G) = P(K) probablity measures on K, where K is the 1-dimensional real projective space n = 2 In any compact affine G-space we can observe: 1-dim real projective G-space or a fixed point (which are tame). for n 2: flag manifolds and their G-quotients. Remark (Veech, Chou G = SL n (R) is Minimally WAP ) Universal semitopological G-compactification G WAP = G { } = 1-point compactification. WAP(SL n (R)) = c 0 (G) R. Remark (In contrast) G Tame is not metrizable.

88 Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Sketch: for n = 2. IA(G) = P(K) probablity measures on K, where K is the 1-dimensional real projective space n = 2 In any compact affine G-space we can observe: 1-dim real projective G-space or a fixed point (which are tame). for n 2: flag manifolds and their G-quotients. Remark (Veech, Chou G = SL n (R) is Minimally WAP ) Universal semitopological G-compactification G WAP = G { } = 1-point compactification. WAP(SL n (R)) = c 0 (G) R. Remark (In contrast) G Tame is not metrizable.

89 Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Sketch: for n = 2. IA(G) = P(K) probablity measures on K, where K is the 1-dimensional real projective space n = 2 In any compact affine G-space we can observe: 1-dim real projective G-space or a fixed point (which are tame). for n 2: flag manifolds and their G-quotients. Remark (Veech, Chou G = SL n (R) is Minimally WAP ) Universal semitopological G-compactification G WAP = G { } = 1-point compactification. WAP(SL n (R)) = c 0 (G) R. Remark (In contrast) G Tame is not metrizable.

90 Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Sketch: for n = 2. IA(G) = P(K) probablity measures on K, where K is the 1-dimensional real projective space n = 2 In any compact affine G-space we can observe: 1-dim real projective G-space or a fixed point (which are tame). for n 2: flag manifolds and their G-quotients. Remark (Veech, Chou G = SL n (R) is Minimally WAP ) Universal semitopological G-compactification G WAP = G { } = 1-point compactification. WAP(SL n (R)) = c 0 (G) R. Remark (In contrast) G Tame is not metrizable.

91 Example G := SL n (R) ( n 2) is nonamenable but conv-int-tame and not int-tame. Sketch: for n = 2. IA(G) = P(K) probablity measures on K, where K is the 1-dimensional real projective space n = 2 In any compact affine G-space we can observe: 1-dim real projective G-space or a fixed point (which are tame). for n 2: flag manifolds and their G-quotients. Remark (Veech, Chou G = SL n (R) is Minimally WAP ) Universal semitopological G-compactification G WAP = G { } = 1-point compactification. WAP(SL n (R)) = c 0 (G) R. Remark (In contrast) G Tame is not metrizable.

92 Example G = S is amenable (hence conv-int-tame) but not int-tame. Example Homeo(C) is not conv-int-tame (C is the Cantor cube).

93 Example G = S is amenable (hence conv-int-tame) but not int-tame. Example Homeo(C) is not conv-int-tame (C is the Cantor cube).

94 Some questions Question Is it true that every (Polish) topological group G is representable on a Rosenthal Banach space? Conjecture: NOT. Candidate Homeo([0, 1] N ) Question Which Polish groups G admit (top.) faithful right topological semigroup compactifications α : G P such that P is (small) : (a) semitopological; (b) metrizable; (c) first-countable? (d) Frechet-Urysohn?

95 Some questions Question Is it true that every (Polish) topological group G is representable on a Rosenthal Banach space? Conjecture: NOT. Candidate Homeo([0, 1] N ) Question Which Polish groups G admit (top.) faithful right topological semigroup compactifications α : G P such that P is (small) : (a) semitopological; (b) metrizable; (c) first-countable? (d) Frechet-Urysohn?

96 Some questions Question Is it true that every (Polish) topological group G is representable on a Rosenthal Banach space? Conjecture: NOT. Candidate Homeo([0, 1] N ) Question Which Polish groups G admit (top.) faithful right topological semigroup compactifications α : G P such that P is (small) : (a) semitopological; (b) metrizable; (c) first-countable? (d) Frechet-Urysohn?

97 Question (Besides SL n (R)) Find more locally compact groups G which are conv-int-tame but nonamenable. Question What if G is discrete? Is it true that there exists a nonamenable but conv-int-tame DISCRETE group?

98 Question (Besides SL n (R)) Find more locally compact groups G which are conv-int-tame but nonamenable. Question What if G is discrete? Is it true that there exists a nonamenable but conv-int-tame DISCRETE group?

99 Kütos!