Approximate amenability and Charles role in its advancement

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1 Approximate amenability and Charles role in its advancement Fereidoun Ghahramani Department of Mathematics University of Manitoba September 2016 Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

2 Derivations A = a Banach algebra X = a Banach A-bimodule Derivation D : A! X is a linear mapping that satisfies D(ab) =D(a) b + a D(b), (a, b 2 A). Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

3 D is inner if 9x 2 X such that D(a) =a x x a, (a 2 A), i.e. D = ad x Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

4 amenable Banach algebras A is amenable if every continuous D : A! X is inner, for all X. Concept founded by Barry Johnson, in late 1960 s. Motivated by his attempts in answering the following question: Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

5 Suppose that G is a locally compact group, L 1 (G) is the group algebra and M(G) is the measure algebra of G. Given µ 2 M(G), the mapping is a derivation. D µ : f 7! f? µ µ? f, (f 2 L 1 (G)) Question Is every derivation D on L 1 (G) a D µ for some µ 2 M(G)? (Barry Johnson s Ph.D. thesis, Cambridge 1962). Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

6 contractible Banach algebras Defn.(A. Helemskii) A is contractible if every continuous D! X is inner, for all X. It is not known whether there exists an infinite-dimensional contractible Banach algebra. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

7 approximately inner derivation Continuous derivation D : A! X is approximately inner if D(a) =lim i ad xi (a), (a 2 A). All derivations will be assumed to be continuous. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

8 Approximately amenable Banach algebra Defn. (R.J. Loy, F. Gh.) A is approximately amenable if every D : A! X is approximately inner for all X. Boundedly approximately amenable: D = lim ad x i (SO), for some operator-norm bounded net (ad x i ). Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

9 Approximately contractible D : A! X,... Boundedly approximating contractible... Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

10 Uniform variants uniformly approximating amenable (contractible): Approximation is in operator-norm topology - instead of strong-operator topology. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

11 Weak approximate amenability Approximation is in weak -topology. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

12 Equivalences 1. (B. E. Johnson, 2000) uniformly approx. contractible, contractible. 2. (A. Pirkowskii, and independenly, R. J. Loy, Y. Zhang, F. Gh., 2005) unif. approx. amen., amen. 3. (R. J. Loy, Y. Zhang, F. Gh., 2004) Approx. amen., Approx. contract., Weak -approx. amen. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

13 Intrinsic characterization of amenability Barry Johnson A is amenable iff there exists a bounded net (M i ) A ˆ A such that: 1 a M i M i a! 0, 2 (M i ) a! a, (a 2 A), Where : A ˆ A! A is the map specified by (a b) =ab, (a, b 2 A). Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

14 Interinsic characterization of approximate amenability A is approximately amenable iff there exists in net (M i ) (A ] ˆ A ] ) such that a M i M i a! 0, (M i )=1, 8i. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

15 (Loy, Gh., 2001) A is approx. amen. if and only if, there exist nets (M i ) (A ˆ A), (F i ), (G i ) A, such that (i) a.m i M i.a + F a a G! 0 (a 2 A) (ii) a.f i! a, G i.a! a (a 2 A) (iii) (M i )=F i + G i 8i. (a b) =ab Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

16 For bounded approximate amenability, additionally, 9C > 0 (i 0 ) ka.m i M i.a + F i a a G i kappleckak, 8a, 8i; (ii 0 ) ka.f i kappleckak, kg i.ak appleckak 8a 2 A, 8i. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

17 For bounded approximate contractibility (F i ), (G i ) A, (M i ) A ˆ A, 9C > 0 k.k appleckak. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

18 Approximate identities 1. (R. J. Loy, F. Gh., 2000) A approx. amen ) A has a right approx. identity and a left approx. identity. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

19 2. (Y. Choi, Y. Zhang, F. Gh., 2007) Suppose that A is bddly. approx. amen. and has a multiplier-bounded right approx. identity and a multiplier-bounded left approx. identity. ) A has a bounded approximate identity. [Defn: (e i ) is multiplier-bounded if 9K > 0, such that kae i kapplek kak, 8a 2 A, 8i]. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

20 (Y. Choi, Y. Zhang, F. Gh., 2007) Suppose that A is boundedly approximately contractible. Then A has a bounded approximate identity. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

21 Does the result extend to bddly. approx. amen. algebras? (C. J. Read, F. Gh., 2010) There exists a boundedly approximately amenable Banach algebra, that has a bounded left approximate identity but no bounded right approximate identity. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

22 Some details: K (l 1 )=compact operators on l 1 K (l 1 ) is known to be amenable (Barry Johnson) (e k )=the standard basis for l 1. For N = 1, 2,...,let T N = kt kop + N lim sup kte k k, (T 2 K (l 1 )) k!1 A N = K (l 1 ),. N Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

23 Detour Defn.: c R (A) =inf{m : A has a bounded right approximate identity of bound M}. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

24 P n = projection onto span{e 1, e 2,...,e n }, n = 1, 2,... (P n ) is a bounded left approximate identity for each A N of bound 1, but c R (A N ) N + 1. So if we set A = c 0 N A N A has a bounded left approximate identity of bound 1, but has no bounded right approximate identity. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

25 And the following is needed: (C. J. Read, F. Gh., 2010). Suppose that (B n ) is a sequence of amenable Banach algebras, and there is an M > 0 such that each B n has a bounded left approximate identity of bound M. Then B = c 0 n B n is boundedly approximately amenable. So Bdd. approx. amen. is not bdd. approx. contract. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

26 The algebra A defined as above has the property : A approximately amenable. A op is not Whereas A ] A op is approximately amenable. So an approximately amenable algebra may contain an ideal of co-dimension 1, that is not approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

27 How about bdd. approx. amen. compared to approx. amen? (C. J. Read, F. Gh., 2013). There exists an approximately amenable Banach algebra which is not boundedly approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

28 Ideas of proof Step 1. Let A be a Banach algebra, (e i ) i2i a bounded left approximate identity for A, and (f j ) j2j a r.a.i for A (not necessarily bounded). Suppose that for every i 2 I and j 2 J there exists d i,j 2 A ˆ A with (d i,j )=e i + f j f j e i, ( (a 1 a 2 )=a 1 a 2 ) and for each a 2 A we have lim j lim sup i ka.d i,j d i,j.ak = 0. Then A is approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

29 Step 2. Suppose that (A n ) 1 n=1 are Banach algebras each of which satisfies the conditions of Step 1, and suppose further that the norms of the b.l.a.i s involved are uniformly bounded. Then the c 0 -direct sum n A n is approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

30 Detour Defn. The bounded approximate amenability constant (baac) of a boundedly approximately amenable Banach algebra is the infimum of all K, such that ka.m i M i.ak applek kak, (a 2 A) extended over all (M i ) (A # ˆ A # ) acting as an approximate diagonal for A. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

31 We find a sequence (A n ) of boundedly approximately amenable Banach algebras that satisfies baac(a n ) n 2, and the sequence (A n) satisfies the hypothesis of the above corollary. Then A = c 0 n A n is the desired algebra. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

32 Open question Is there an approximately amenable Banach algebra with no two-sided approximate identity? Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

33 Approximate amenability in abstract harmonic analysis G = a locally compact group L 1 (G) = the group algebra of G M(G) = the measure algebra of G Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

34 (R. J. Loy, F. Gh., 2000) (i) L 1 (G) is approximately amenable if and only if G is amenable. (ii) M(G) is approximately amenable if and only if G is discrete and amenable. (Using a result of H. Dales, A. Helemeskii, F. Gh., 2000) (iii) L 1 (G) (with 1st or 2nd Arens product) is approximately amenable if and only if G is finite. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

35 The Fourier algebra A(G) A(G) ={f ǧ : f, g 2 L 2 (G)} ǧ(x) =g(x 1 ) (x 2 G) And for h 2 A(G), khk A(G) = inf{kf k 2 kgk 2 : h = f ǧ, f 2 L 2 (G), g 2 L 2 (G)} Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

36 (B. Forrest, V. Runde, 2004) A(G) is amenable if and only if G has an amenable subgroup of finite index. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

37 (R. Stokke, F. Gh., 2005) Suppose that G is amenable and contains an open abelian subgroup. Then A(G) is approximately amenable. (Y. Choi, F. Gh., 2009) Suppose that G has an open abelian subgroup. Then A(G) is boundedly approximately amenable, if and only if G is amenable. Corollary. Suppose that G is discrete and amenable. Then A(G) is approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

38 Example Let >< H = 6 4 >: 1 m n 0 1 p >= : m, n, p 2 Z ; >; the Heisenberg group. A(H) is approx. amen., but not amen. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

39 Open question Suppose that G is a discrete group. Does approximate amenability of A(G) ) amenability of G? (Y. Choi, Y. Zhang, F. Gh., 2007) If F 2 is a closed subgroup of G, then A(G) is not approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

40 (Y. Choi, F. Gh, 2010) A(G) (with an Arens product) is boundedly approximately amenable if and only if G is finite. This in particular extends a result of E. Granirer, who proved that A(G) is amenable if and only if G is finite Open question: When is A(G) approximately amenable? Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

41 Beurling algebras weight: w continuous. w : G! (0, 1) w(xy) apple w(x)w(y) (x, y 2 G) L 1 (G, w) ={f : fw 2 L 1 (G)} kf k w = kwf k 1 Z (f g)(x) = f (xy)g(y 1 ) d(y), G (f, g 2 L 1 (G, w), a.e.x). Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

42 Amenability of Beurling algebras (Niels Gronbaek 1990). Suppose that w(e) =1. Then L 1 (G, w) is amenable if and only if (i) G is amenable; (ii) x 7! w(x)w(x 1 ) is bounded on G (w is diagonally bounded). An alternative proof of this result in general case (not assuming w(e) =1) was given by (R. loy, Y. Zhang and F. Gh), in Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

43 Defn. A weight on G is symmetric if w(x 1 )=w(x) 8x 2 G. (E. Samei, Y. Zhang, F. Gh., 2008) Suppose that w is symmetric. Then L 1 (G, w) is boundedly approximately contractible if and only if G is amenable and w is bounded. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

44 Segal algebras (S(G), k.k S ) is a Segal subalgebra of L 1 (G) if: (i) S(G) is a dense linear subspace of L 1 (G); (ii) k.k S k.k 1 ; (iii) S(G) is left translation invariant and x 7! l x f : G! S(G) is continuous; (iv) kl x f k S = kf k S (f 2 S(G)). S(G) is symmetric if, additionally, the above conditions hold on the right. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

45 Example S 1,p = L 1 (G) \ L p (G) kf k = kf k 1 + kf k p (f 2 S 1,p ) and convolution product. If G is unimodular, S 1,p is a symmetric Segal algebra. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

46 (Y. Choi, Y. Zhang, F. Gh., 2007) A proper symmetric Segal subalgebra of L 1 (G) can never be boundedly approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

47 (R. Loy, H.G. Dales, 2010) Specific Segal algebras on R n, T n are not approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

48 (Y. Choi, F. Gh., 2009) No non-trivial Segal subalgebra of R n or T n is approximately amenable. (M. Alagamandan, 2015) If G is a SIN group, then no non-trivial symmetric Segal algebra S(G) can be approximately amenable. Conjecture: Non-trivial Segal algebras are never approximately amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

49 Operator algebras Open question: When is a C - algebra approximately amenable? (Y. Choi, Y. Zhang, F. Gh., 2008) Let be a discrete group. Then the following are equivalent: (i) The full group C -algebra C ( ) of is approximately amenable. (ii) The reduced C -algebra C r ( ) is approximately amenable. (iii) is amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

50 non-self adjoint operator algebras (Y. Choi, and independently, C. J. Read, F. Ghahramani). There exists a non-self-adjoint operator algebra which is approximately amenable but not amenable. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

51 Approximate amenability of K (X) K (X) =Compact operators on X Amenability of K (X), for X = C( ), where is uncountable, compcat metrizable space and X = l p, 1 < p < 1 was established by Barry Johnson, Mem. AMS, Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

52 (N. Grønbaek, B. E. Johnson, G. A. Willis, 1994). Suppose that X = X 1 X 2 and K (X) is amenable. Then at least one of the maps or 1 : K (X 1, X) ˆ K (X, X 1 )! K (X) is surjective. 1 (T 1 T 2 )=T 1 T 2 (T 1 2 K (X 1, X), T 2 2 K (X, X 1 )) 2 : K (X 2, X) ˆ K (X, X 2 )! K (X) (1) 2 (T 1 T 2 )=T 1 T 2, (T 1 2 K (X 2, K ), T 2 2 K (X, X 2 )) Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

53 A question which was open since 2000: Is there a Banach space X for which K (X) is approximately amenable, but not amenable? Yes, C. J. Read, F. Gh. (2014). Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54

54 Theorem If X is fairly close to Hilbert space, then K (X) is approximately amenable. Then, we define X 1 and X 2 such that X = X 1 X 2 fails the [Grønback-Johnson-Willis] necessary condition for amenability of K (X), and yet X is fairly close to Hilbert space. Fereidoun Ghahramani (Universitiy of Manitoba) Approximate amenability and Charles role in its advancement September / 54