Algebraic Methods in Functional Analysis: The Victor Shulman by Ivan G. Todorov, Lyudmila Turowska

By Ivan G. Todorov, Lyudmila Turowska

This quantity contains the court cases of the convention on Operator concept and its purposes held in Gothenburg, Sweden, April 26-29, 2011. The convention was once held in honour of Professor Victor Shulman at the social gathering of his sixty fifth birthday. The papers integrated within the quantity conceal a wide number of subject matters, between them the idea of operator beliefs, linear preservers, C*-algebras, invariant subspaces, non-commutative harmonic research, and quantum teams, and mirror contemporary advancements in those components. The e-book involves either unique study papers and top of the range survey articles, all of that have been rigorously refereed. ​

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Extra info for Algebraic Methods in Functional Analysis: The Victor Shulman Anniversary Volume

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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. [16] 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).

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