Amenability properties of the non-commutative Schwartz space

Transcription

1 Amenability properties of the non-commutative Schwartz space Krzysztof Piszczek Adam Mickiewicz University Poznań, Poland Workshop on OS, LCQ Groups and Amenability, Toronto, Canada, May, 2014.

2 Plan 1 Definition of the non-commutative Schwartz space

3 Plan 1 Definition of the non-commutative Schwartz space 2 Known results - short and informative

4 Plan 1 Definition of the non-commutative Schwartz space 2 Known results - short and informative 3 Amenability

5 The space S s = {ξ + = (ξ j ) j N C N : ξ 2 k := ξ j 2 j 2k < + for all k N 0 }, j=1

6 The space S s = {ξ + = (ξ j ) j N C N : ξ 2 k := ξ j 2 j 2k < + for all k N 0 }, s C ([a, b]), j=1

7 The space S s = {ξ + = (ξ j ) j N C N : ξ 2 k := ξ j 2 j 2k < + for all k N 0 }, j=1 s C ([a, b]), s S(R),

8 The space S s = {ξ + = (ξ j ) j N C N : ξ 2 k := ξ j 2 j 2k < + for all k N 0 }, j=1 s C ([a, b]), s S(R), { + s = η = (η j ) j N C N : η 2 k := } η j 2 j 2k < + for some k N 0 j=1

9 The space S s = {ξ + = (ξ j ) j N C N : ξ 2 k := ξ j 2 j 2k < + for all k N 0 }, j=1 s C ([a, b]), s S(R), { + s = η = (η j ) j N C N : η 2 k := } η j 2 j 2k < + for some k N 0 j=1 L(s, s) := {linear and continuous maps x : s s}, topology on L(s, s) given by x k := sup{ xξ k : ξ k 1}.

10 The algebra S The map ι: s s is continuous.

11 The algebra S The map ι: s s is continuous. For x, y L(s, s) we define x y := x ι y.

12 The algebra S The map ι: s s is continuous. For x, y L(s, s) we define x y := x ι y. Since xy k x k y k, L(s, s) is lmc.

13 The algebra S The map ι: s s is continuous. For x, y L(s, s) we define x y := x ι y. Since xy k x k y k, L(s, s) is lmc. By duality ξ, η := + j=1 ξ j η j ξ s, η s we have

14 The algebra S The map ι: s s is continuous. For x, y L(s, s) we define x y := x ι y. Since xy k x k y k, L(s, s) is lmc. By duality ξ, η := + j=1 ξ j η j ξ s, η s we have x ξ, η := ξ, xη, ξ, η s, x L(s, s).

15 The algebra S The map ι: s s is continuous. For x, y L(s, s) we define x y := x ι y. Since xy k x k y k, L(s, s) is lmc. By duality ξ, η := + j=1 ξ j η j ξ s, η s we have x ξ, η := ξ, xη, ξ, η s, x L(s, s). Definition S := (L(s, s),, ).

16 The name Non-commutative Schwartz space because S = s ε s s S(R).

17 The name Non-commutative Schwartz space because S = s ε s s S(R). Algebra of smooth operators because S C ([a, b]).

18 Importance of S 1 Structure theory of Fréchet spaces: nuclearity, splitting of short exact sequences.

19 Importance of S 1 Structure theory of Fréchet spaces: nuclearity, splitting of short exact sequences. 2 K-theory the work of Cuntz, Phillips and others.

20 Importance of S 1 Structure theory of Fréchet spaces: nuclearity, splitting of short exact sequences. 2 K-theory the work of Cuntz, Phillips and others. 3 Connection to C -dynamical systems.

21 Importance of S 1 Structure theory of Fréchet spaces: nuclearity, splitting of short exact sequences. 2 K-theory the work of Cuntz, Phillips and others. 3 Connection to C -dynamical systems. 4 Locally convex operator spaces work of Effros et al.

22 Properties of S 1) S is an algebra of matrices. Indeed, S B(l 2 ) continuously (in fact S S p (l 2 )), reason: l 2 s, s l 2.

23 Properties of S 1) S is an algebra of matrices. Indeed, S B(l 2 ) continuously (in fact S S p (l 2 )), reason: l 2 s, s l 2. x = ( xe j, e i ) i,j = (x ij ) i,j S x k, := sup x ij (ij) k < + for all k N. i,j

24 Properties of S 1) S is an algebra of matrices. Indeed, S B(l 2 ) continuously (in fact S S p (l 2 )), reason: l 2 s, s l 2. x = ( xe j, e i ) i,j = (x ij ) i,j S x k, := sup x ij (ij) k < + for all k N. i,j Proposition (Bost, 1990; independently Domański, 2012) If S 1 := S C then σ S1 (x) = σ B(l2 )(x).

25 Properties of S 1) S is an algebra of matrices. Indeed, S B(l 2 ) continuously (in fact S S p (l 2 )), reason: l 2 s, s l 2. x = ( xe j, e i ) i,j = (x ij ) i,j S x k, := sup x ij (ij) k < + for all k N. i,j Proposition (Bost, 1990; independently Domański, 2012) If S 1 := S C then σ S1 (x) = σ B(l2 )(x). 2) Approximate identities Definition (u α ) α Λ is an a.i. if u α x x and xu α x for all x. It is bounded if the set of u α s is bounded. It is sequential if Λ = N.

26 Properties of S 1) S is an algebra of matrices. Indeed, S B(l 2 ) continuously (in fact S S p (l 2 )), reason: l 2 s, s l 2. x = ( xe j, e i ) i,j = (x ij ) i,j S x k, := sup x ij (ij) k < + for all k N. i,j Proposition (Bost, 1990; independently Domański, 2012) If S 1 := S C then σ S1 (x) = σ B(l2 )(x). 2) Approximate identities Definition (u α ) α Λ is an a.i. if u α x x and xu α x for all x. It is bounded if the set of u α s is bounded. It is sequential if Λ = N. Proposition ( ) In 0 If u n := then (u 0 0 n ) n is a sequential a.i. No b.a.i. in S.

27 The work of Ciaś 1 Functional calculus.

28 The work of Ciaś 1 Functional calculus. 2 Characterization of closed, commutative -subalgebras of S.

29 The work of Ciaś 1 Functional calculus. 2 Characterization of closed, commutative -subalgebras of S. 3 Some consequences: (i) S = span S +. (ii) x 0 x = y y for some y S xξ, ξ 0 ξ s.

30 Automatic continuity results Theorem 1 All positive functionals on S are automatically continuous.

31 Automatic continuity results Theorem 1 All positive functionals on S are automatically continuous. 2 Every derivation into any S-bimodule is automatically continuous.

32 Automatic continuity results Theorem 1 All positive functionals on S are automatically continuous. 2 Every derivation into any S-bimodule is automatically continuous. Important in all automatic continuity proofs: S is nuclear, i.e. S π S = S ε S, equivalently, unconditionally summable sequences are absolutely summable. Consequently, bounded subsets are relatively compact.

33 Amenability of groups Definition A locally compact group G is amenable if there exists an invariant mean on L (G).

34 Amenability of groups Definition A locally compact group G is amenable if there exists an invariant mean on L (G). Observation Amenable are all: 1 compact groups Haar measure is finite, 2 abelian groups use Markov-Kakutani fixed point theorem.

35 Amenability of groups Definition A locally compact group G is amenable if there exists an invariant mean on L (G). Observation Amenable are all: 1 compact groups Haar measure is finite, 2 abelian groups use Markov-Kakutani fixed point theorem. Theorem (B.E. Johnson, 1972) TFAE for a locally compact group G: 1 G is amenable, 2 for every L 1 (G)-bimodule X and every continuous derivation δ : L 1 (G) X there is an x X with δ(a) = a x x a.

36 Amenability of algebras Definition A (Banach, Fréchet,...) algebra A is amenable if for every (Banach, Fréchet,...) A-bimodule X any continuous derivation D : A X is inner.

37 Amenability of algebras Definition A (Banach, Fréchet,...) algebra A is super-amenable if for every (Banach, Fréchet,...) A-bimodule X any continuous derivation D : A X is inner.

38 Amenability of algebras Definition A (Banach, Fréchet,...) algebra A is super-amenable if for every (Banach, Fréchet,...) A-bimodule X any continuous derivation D : A X is inner. Fundamental Question When a continuous derivation is inner?

39 Amenability of algebras A bunch of results. 1 Selivanov, 1976: If A (AP) is super-amenable then A = M n1... M nk.

40 Amenability of algebras A bunch of results. 1 Selivanov, 1976: If A (AP) is super-amenable then A = M n1... M nk. 2 Selivanov, 1976: L 1 (G) is super-amenable iff G is finite.

41 Amenability of algebras A bunch of results. 1 Selivanov, 1976: If A (AP) is super-amenable then A = M n1... M nk. 2 Selivanov, 1976: L 1 (G) is super-amenable iff G is finite. 3 Helemskii: The only super-amenable and commutative Banach algebras are C n.

42 Amenability of algebras A bunch of results. 1 Selivanov, 1976: If A (AP) is super-amenable then A = M n1... M nk. 2 Selivanov, 1976: L 1 (G) is super-amenable iff G is finite. 3 Helemskii: The only super-amenable and commutative Banach algebras are C n. 4 Connes, 1978; Haagerup, 1983: A C -algebra is amenable if and only if it is nuclear.

43 Theorem (A. Pirkovskii, 2009) The non-commutative Schwartz space is not amenable.

44 Theorem (A. Pirkovskii, 2009) The non-commutative Schwartz space is not amenable. Question Due to (weaker) notions of approximate amenability, find out how approximately amenable our algebra is?

45 Theorem (A. Pirkovskii, 2009) The non-commutative Schwartz space is not amenable. Question Due to (weaker) notions of approximate amenability, find out how approximately amenable our algebra is? Definition A derivation δ : A X is uniformly approximately inner if δ(a) = lim α (a x α x α a) a A and (x α ) α is bounded in X. A is uniformly approximately amenable if every continuous derivation into any dual A-bimodule is uniformly approximately inner.

46 Theorem (A. Pirkovskii, 2009) The non-commutative Schwartz space is not amenable. Question Due to (weaker) notions of approximate amenability, find out how approximately amenable our algebra is? Definition A derivation δ : A X is boundedly approximately inner if δ(a) = lim α (a x α x α a) a A and (a a x α x α a) α is equicontinuous in L(A, X ). A is boundedly approximately amenable if every continuous derivation into any dual A-bimodule is boundedly approximately inner.

47 Theorem (A. Pirkovskii, 2009) The non-commutative Schwartz space is not amenable. Question Due to (weaker) notions of approximate amenability, find out how approximately amenable our algebra is? Definition A derivation δ : A X is approximately inner if δ(a) = lim(a x α x α a) a A α and (x α ) α X. A is approximately amenable if every continuous derivation into any dual A-bimodule is approximately inner.

48 S is not boundedly approximately amenable Definition A a Fréchet lmc algebra. An approximate identity (u α ) α A is called a multiplier-bounded left approximate identity if the set {u α a: α} is bounded for every a A.

49 S is not boundedly approximately amenable Definition A a Fréchet lmc algebra. An approximate identity (u α ) α A is called a multiplier-bounded right approximate identity if the set {au α : α} is bounded for every a A.

50 S is not boundedly approximately amenable Definition A a Fréchet lmc algebra. An approximate identity (u α ) α A is called a multiplier-bounded approximate identity if the set {u α a, au α : α} is bounded for every a A.

51 S is not boundedly approximately amenable Definition A a Fréchet lmc algebra. An approximate identity (u α ) α A is called a multiplier-bounded approximate identity if the set {u α a, au α : α} is bounded for every a A. Theorem (Choi, Ghahramani, Zhang, 2009) If a Fréchet lmc algebra is boundedly approximately amenable and possesses both, multiplier bounded left and right approximate identity then it necessarily admits a bounded approximate identity.

52 S is not boundedly approximately amenable Definition A a Fréchet lmc algebra. An approximate identity (u α ) α A is called a multiplier-bounded approximate identity if the set {u α a, au α : α} is bounded for every a A. Theorem (Choi, Ghahramani, Zhang, 2009) If a Fréchet lmc algebra is boundedly approximately amenable and possesses both, multiplier bounded left and right approximate identity then it necessarily admits a bounded approximate identity. Corollary The non-commutative Schwartz space is not ( boundedly ) In 0 approximately amenable. (Because u n := is a m.b.a.i.) 0 0

53 S is approximately amenable Theorem (Dales, Loy, Zhang, 2006) A Fréchet lmc algebra (A, ( n ) n N ) is approximately amenable if and only if for each ε > 0, each finite subset S of A and every k N there exist F A A and u, v A such that π(f ) = u + v and for each a S: (i) a F F a + u a a v k < ε, (ii) a au k < ε and a va k < ε.

54 S is approximately amenable Theorem (Dales, Loy, Zhang, 2006) A Fréchet lmc algebra (A, ( n ) n N ) is approximately amenable if and only if for each ε > 0, each finite subset S of A and every k N there exist F A A and u, v A such that π(f ) = u + v and for each a S: (i) a F F a + u a a v k < ε, (ii) a au k < ε and a va k < ε. Recall: π(a b) := ab, a (x y) := ax y, (x y) b := x yb.

55 S is approximately amenable Theorem (Dales, Loy, Zhang, 2006) A Fréchet lmc algebra (A, ( n ) n N ) is approximately amenable if and only if for each ε > 0, each finite subset S of A and every k N there exist F A A and u, v A such that π(f ) = u + v and for each a S: (i) a F F a + u a a v k < ε, (ii) a au k < ε and a va k < ε. Recall: π(a b) := ab, a (x y) := ax y, (x y) b := x yb. Simplifications for S : S S = S(S) is just matrices of matrices and

56 S is approximately amenable Theorem (Dales, Loy, Zhang, 2006) A Fréchet lmc algebra (A, ( n ) n N ) is approximately amenable if and only if for each ε > 0, each finite subset S of A and every k N there exist F A A and u, v A such that π(f ) = u + v and for each a S: (i) a F F a + u a a v k < ε, (ii) a au k < ε and a va k < ε. Recall: π(a b) := ab, a (x y) := ax y, (x y) b := x yb. Simplifications for S : S S = S(S) is just matrices of matrices and it is enough to work with singletons.

57 S is approximately amenable Theorem The non-commutative Schwartz space is approximately amenable.

58 S is approximately amenable Theorem The non-commutative Schwartz space is approximately amenable. Proof. For any a S and u = v = u n and F = diag(u n,..., u n, 0, 0,...): a F F a + u a a v k < ε, a au k < ε, a va k < ε.

59 S is approximately amenable Theorem The non-commutative Schwartz space is approximately amenable. Proof. For any a S and u = v = u n and F = diag(u n,..., u n, 0, 0,...): a F F a + u a a v k < ε, a au k < ε, a va k < ε. The problem is π(f ) = u n 2u n = u + v and we need to slightly perturb the big matrix F.

60 S is approximately amenable Theorem The non-commutative Schwartz space is approximately amenable. Proof. For any a S and u = v = u n and F = diag(u n,..., u n, 0, 0,...): a F F a + u a a v k < ε, a au k < ε, a va k < ε. The problem is π(f ) = u n 2u n = u + v and we need to slightly perturb the big matrix F. The solution is u n + 1 n e 1 11 n e n e n n e 12 u n + 1 n e n e n F = n e 1 1n n e 2n... u n + 1 n e nn