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Extra info for Algebraic Methods in Functional Analysis: The Victor Shulman Anniversary Volume
If ???? is biﬂat then the continuous Hochschild cohomology groups ℋ???? (????, ????∗ ) vanish for all ???? ≥ 1. In particular, biﬂat algebras are weakly amenable. Let ???? denote the Banach algebra obtained by equipping ℓ1 with pointwise multiplication. ???? is a standard example of a commutative, semisimple, biﬂat Banach algebra that has no bounded approximate identity, and hence is nonamenable. 1, it is singly generated as a Banach algebra. It turns out that there is a continuous algebra homomorphism ???? : ???? → ℬ(ℋ) whose range is closed, so that ????(????) is a singly generated, biﬂat operator algebra.
Camb. Phil. , 142 (2007), pp. 111–123. A. Gifford, Operator algebras with a reduction property, J. Aust. Math. , 80 (2006), pp. 297–315. Ya. Helemski˘ı, The Homology of Banach and Topological Algebras, vol. 41 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1989. E. , 1972. W. Marcoux, On abelian, triangularizable, total reduction algebras, J. Lond. Math. Soc. (2), 77 (2008), pp. 164–182.  J. Peterson, personal communication via MathOverﬂow.
Proof. Regard ???? as a closed subalgebra of some ℬ(ℋ). Since the unitization of an amenable Banach algebra is amenable, we may assume without loss of generality that ???? contains the identity operator ????. ˆ???? → Let (Δ???? ) be a bounded approximate diagonal for ????. Deﬁne ???? : ???? ⊗ ℬ(ℬ(ℋ)) by ???? (???? ⊗ ????)(????) = ????????????, and let ???? be a point-to-weak∗ cluster point of the net ???? (Δ???? ) ⊂ ℬ(ℬ(ℋ)). As (Δ???? ) is a bounded approximate diagonal for ????, it follows from routine estimates and convergence arguments that the following properties hold: (i) ????(????) ∈ ????′ for all ???? ∈ ℬ(ℋ); (ii) ????(????) = ???? for all ???? ∈ ????′ ; (iii) ????(????????????) = ????????(????)???? for all ????, ???? ∈ ????′ and all ???? ∈ ℬ(ℋ).