Reflexivity and hyperreflexivity of bounded n-cocycle spaces and application to convolution operators. İstanbul Analysis Seminars, İstanbul, Turkey

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1 Reflexivity and hyperreflexivity of bounded n-cocycle spaces and application to convolution operators İstanbul Analysis Seminars, İstanbul, Turkey Ebrahim Samei University of Saskatchewan A joint work with Jafar Soltani Farsani March 18, 2016

2 Let X be a Banach space, and let A B(X) be an algebra of bounded operators on X with unit (i.e. id X A). (1) LatA denotes the set of all closed subspaces of X invariant under A, i.e., LatA := {I X : T (I) I for all T A}. (2) the algebra generated by LatA is defined to be the set of all T B(X) such that T (I) I for each I LatA; This is denoted by alglata. (3) Clearly A alglata. We say that A is reflexive if alglata = A.

3 Examples: (i) von Neumann algebras: It follows easily from the double-commutate Theorem. (ii) Weakly closed unital algebras of normal operators (Sarason). (iii) operator algebras generated by commutative subspace lattices or briefly CSL algebras (Arveson).

4 Invariant subspace problem and reflexivity: Let X be a Banach space with dimx > 1, T B(X) and defined A = alg{id X, T } w.o.t, where w.o.t stands for the weak operator topology. If A is reflexive, then T has a non-trivial invariant subspace.

5 Let X and Y be Banach spaces, and B(X, Y ) be the spaces of bounded (linear) operators from X into Y. Let S be a linear subspace of S(X, Y ). Put ref(s) = {T B(X, Y ) x X, {S n } S : T (x) = lim n S n (x)} = {T B(X, Y ) x X, T (x) S(x)}. Clearly S ref(s). We say that S is reflexive if S = ref(s).

6 If X = Y = H, a Hilbert space, then ref(s) = {T B(H) Q T P = 0, P, Q B(H) projections with Q SP = 0}. (Larson-Kraus) A weak -closed subspace S of B(H) is reflexive if and only if its preannihilator, S, is the closed span of rank-1 operators in S.

7 Let A be a Banach algebra, let X be a Banach A-bimodule, and let D : A X be an operator. (i) D is a derivation if for all a, b A, D(ab) = ad(b) + D(a)b. (ii) D is a local derivation if for each a A, there is a derivation D a : A X such that D(a) = D a (a). If, in addition, D is bounded, we say that D is a bounded local derivation. Question 1: (Kadison) For what A and X (bounded) local derivations are derivations? Let Z 1 (A, X) be the linear spaces of bounded derivations from A into X. Question 2: (Larson) For what A and X, Z 1 (A, X) is reflexive?

8 Bounded local derivations are derivations in the following cases: (R. Kadison 1990) A is a von Neumann algebra and X is a dual module. (D. Larson and A. Sourour 1990) A = X = B(E), E is a Banach space. (V. Shulman 1994) A is a C -algebra and X = A. (B. E. Johnson 2000) A is a C -algebra. (E. Samei 2005, 2011) A = L 1 (G) for G having an open subgroup with polynomial growth (eg. IN groups, totally disconnected groups).

9 Definition 1 (Alaminos-Extremera-Villena) A Banach algebra A has the property (B) if for any T (Aˆ A) satisfying T (a b) = 0 for all a, b A with ab = 0, we have T (ab c) = T (a bc) (a, b, c A). Note: If A has a b.a.i, then A has the property (B) if and only if for m : A A A, a b ab, ker m = span{a b a, b A, ab = 0}. Theorem 2 (Alaminos-Extremera-Villena) C -algebras and group algebras have the strong property (B).

10 Let X and Y be Banach spaces, and let S be a linear subspace of B(X, Y ). For every T B(X, Y ), we define and dist(t, S) = inf S S = inf S S T S sup T (x) S(x). x 1 dist r (T, S) = sup x 1 inf S S T (x) S(x). Clearly dist r (T, S) dist(t, S). We say that S is hyperreflexive if there exists C := C S > 0 such that dist(t, S) Cdist r (T, S) for all T B(X, Y ). The smallest possible value for C is called the hyperreflexivity constant of S.

11 If X = Y = H, a Hilbert space, then dist r (T, S) = sup{ Q T P P, Q B(H) projections with Q SP = 0}. If, in addition, S is a unital algebra, then dist r (T, S) = sup{ P T P P Lat S}. (Arveson) Let S be a reflexive subspace. Then S is hyperreflexive if and only if every element in the preannihilates of S is an l 1 -sum of rankone tensors that annihilate S.

12 Examples: (Arveson) Nest algebras with C = 1. (Christensen) Injective von Neumann algebras with C 4. (Rosenoer, Larson-Kraus) any weak -closed subspace of normal operators with C 3. (Alaminos-Extremera-Villena) Z 1 (L 1 (G), L 1 (G)) if G has an open subgroup with polynomial growth. (Samei-Soltani) Z 1 (L 1 (G), X) if G has an open subgroup with polynomial growth and X is a Banach L 1 (G)-bimodule with H 2 (L 1 (G), X) a Banach Space. Question: What can we say about the hyperreflexivity constant of derivation spaces?

13 Alaminos-Extremera-Villena: Let A be either a C -algebra or a group algebra. Then there is a continuous function L : [0, 1) [0, ) with L(0) = 0 such that for any continuous bilinear map T : A A C with T 1, we have T (ab, c) T (a, bc) L(α) (a, b, c A 1 ), where 0 α < 1 with sup{ T (a, b) : a, b A 1, ab = 0} α. Question: Can L be chosen to be linear?

14 Definition 3 We say that A has the strong property (B) with constant C > 0 if for any T (Aˆ A), T (ab c) T (a bc) Cα(T ) (a, b, c A 1 ), where α(t ) = sup{ T (a b) : a, b A 1, ab = 0}. Note: For m : A A A, a b ab and I := ker m, α(t ) ResT I. If A has the strong property (B) with constant C and an approximate identity bounded by M, then ResT I 2CMα(T ).

15 Fourier algebra of the unit circle: T = {z C : z = 1}. A(T) = l 1 (Z); ˆf(n) = 1 2π f = n Z ˆf(n). 2π 0 f(t)e intπ dt A(T) is a Banach algebra of continuous functions on T. Proposition 4 Let T be the unit circle. Then for any T A(T T) and a, b, c A(T) 1, we have T (ab c) T (a bc) 288π(1 + 2)α(T ), where α(t ) = sup{ T (a b) : supp a supp b = }. Theorem 5 Let A be a C -algebra or a group algebra. Then A has the strong property (B) with C = 288π(1 + 2).

16 Let A be a Banach algebra, and X a Banach A-bimodule. The connecting map is defined to be δ 0 := δ : X B(A, X) x δ x, δ x (a) = ax xa. Each δ x is a derivation; it is called an inner derivation. The first Hochschild cohomology of A with coefficient in X is H 1 (A, X) = Z 1 (A, X)/Im δ.

17 For n N, let A (n) be the Cartesian product of n copies of A, and B n (A, X) the space of bounded n-linear maps from A (n) into X. The connecting maps are defined by δ n : B n (A, X) B n+1 (A, X) δ n (T )(a 1,..., a n+1 ) = a 1 T (a 2,..., a n+1 ) + n j=1 ( 1) j T (a 1,..., a j a j+1,..., a n+1 ) } {{ } jth + ( 1) n+1 T (a 1,..., a n )a n+1. Fact: For all n N {0}, δ n+1 δ n = 0.

18 Elements of Z n (A, X) := ker δ n are called bounded n-cocycles. The n th Hochschild cohomology of A with coefficient in X is H n (A, X) = Z n (A, X)/Im δ n 1. Clearly, H n (A, X) is a vector space. Also it is a Banach space iff Im δ n 1 is closed in B n (A, X) iff there is a constant C satisfying dist(t, Z n 1 (A, X)) C δ n 1 (T ), for all T B n 1 (A, X).

19 Theorem 6 Z 1 (A, X) is hyperreflexive with the constant C(288π(1+ 2)+4) 2 in either of the following cases: (i) A is a nuclear C -algebra and X is a dual module; (ii) X = B(H), H is a Hilbert space and A is an injective von Neumann subalgebra of B(H); (ii) A = L 1 (G) for certain amenable locally compact group G with and X is a dual module. Here C is a constant satisfying dist(t, Z 1 (A, X)) C δ T, (T B(A, X)).

20 Let X and Y be Banach spaces, n N, and S be a linear subspace of B n (X, Y ). For every T B n (X, Y ), we define and dist r (T, S) = sup x i 1 dist(t, S) = inf S S T S inf T (x 1,..., x n ) S(x 1,..., x n ). S S It is clear that for all T B n (X, Y ), dist r (T, S) dist(t, S). We define S to be reflexive if dist r (T, S) = 0 = dist(t, S) = 0. We define S to be hyperreflexive if there exist some C > 0 such that for all T B n (X, Y ), dist(t, S) Cdist r (T, S).

21 Theorem 7 Z n (A, X) is hyperreflexive with the constant C2 n 1 (288π(1+ 2)+4) n+1 in either of the following cases: (i) A is a nuclear C -algebra and X is a dual module; (ii) X = B(H), H is a Hilbert space and A is an injective von Neumann subalgebra of B(H); (ii) A = L 1 (G) for certain amenable locally compact group G with and X is a dual module. Here C is a constant satisfying dist(t, Z n (A, X)) C δ n 1 (T ), (T B n (A, X)).

22 Theorem 8 (Christensen) Let H be a Hilbert space, and let A B(H) be a C -algebra. Then TFAE: (i) H 1 (A, B(H)) is a Banach space; (ii) H 1 (A, B(H)) = {0}; (iii) A, the commutant of A in B(H), is hyperreflexive.

23 Theorem 9 Let A be a Banach algebra having the strong property (B) with the constant C, X a Banach space and π : A B(X) a non-degenarate representation. If A has an a.i. bounded by M and H 1 (A, B(X)) is a Banach space, then π(a) = {T B(X) : T π( ) = π( )T }, is hyperreflexive with the constant C A := MCK, (K is the constant coming from the open mapping theorem).

24 For a locally compact group G and 1 < p <, we recall that an operator T B(L p (G)) is a convolution operator if for every t G and f L p (G), T (δ t f) = δ t T (f). The space of all convolution operators on L p (G) is denoted by CV p (G). It is straightforward to check that CV p (G) is a w -closed subalgebra of B(L p (G)) = (L p (G) L q (G)), where q is the conjugate of p.

25 Corollary 10 Let G be a locally compact group, and let CV p (G) be the space of convolution operators on L p (G) (1 p < ). Then CV p (G) is reflexive. If, in addition, G is amenable, then CV p (G) is hyperreflexive with the constant C = 288π(1 + 2). Proof: Let λ : G B(L p (G)) be the left regular representation: Then λ(t)f = δ t f. CV p (G) = λ(g).