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Dual Banach algebras: representations and injectivity. (English) Zbl 1115.46038
A dual Banach algebra is a Banach algebra which is a dual Banach space such that multiplication is separately continuous in the weak$$^\ast$$ topology. Examples of dual Banach algebras are, among others, $$A^{\ast\ast}$$ for any Arens regular Banach algebra $$A$$, the Banach algebra $${\mathcal B}(E)$$ of all bounded, linear operators on a reflexive Banach space $$E$$, the measure algebra of a locally compact group, and – most prominently – the von Neumann algebras.
In the very impressive paper under review, the author develops the theory of general dual Banach algebras with striking, often surprising parallels to von Neumann algebras.
If $$E$$ is a reflexive Banach space, then $${\mathcal B}(E)$$ is a dual Banach algebra, and so is each of its weak$$^\ast$$ closed subalgebras. The author’s first major result is that these are all dual Banach algebras: if $$A$$ is a dual Banach algebra, then there is a reflexive Banach space $$E$$ and a weak$$^\ast$$-weak$$^\ast$$ continuous, isometric algebra homomorphism from $$A$$ into $${\mathcal B}(E)$$. The proof makes ingenious use of real interpolation as a substitute for the GNS-construction.
One reason why dual Banach algebras have attracted interest is that they are the natural setting for the development Connes-amenability: a variant of Banach algebraic amenability that takes the dual space structure into account [see, for instance, V. Runde, Stud. Math. 148, No. 1, 47–66 (2001; Zbl 1003.46028)]. Connes-amenability is an important (and well understood) property and equivalent to a number of other important von Neumann algebraic properties, such as injectivity [see M. Takesaki, “Theory of operator algebras III” (Encyclopaedia of Mathematical Sciences 127(VIII), Springer–Verlag, Berlin) (2003; Zbl 1059.46032)].
In the present paper, the author defines a dual Banach algebra $$A$$ to be injective if the following holds: if $$E$$ is a reflexive Banach space and $$\pi:A\to{\mathcal B}(E)$$ is a weak$$^\ast$$-weak$$^\ast$$ continuous homomorphism, then there is a bounded projection $${\mathcal Q}:{\mathcal B}(E)\to\pi(A)'$$ – where $$\pi(A)'$$ denotes the commutant of $$\pi(A)$$ in $${\mathcal B}(E)$$ – which is a $$\pi(A)'$$-bimodule homomorphism. (For von Neumann algebras, this is equivalent to the usual notion of injectivity.) He then proves that a dual Banach algebra is Connes-amenable if and only if it is injective. (The proof of the “only if” part is relatively easy, but the “if” part again requires sophisticated interpolation techniques.)
There are various other results, besides those two major ones, that are too technical to be reported here. In particular, the author develops a theory of tensor products for dual Banach algebras.

##### MSC:
 46H05 General theory of topological algebras 46B70 Interpolation between normed linear spaces 46H15 Representations of topological algebras 46M05 Tensor products in functional analysis 47L10 Algebras of operators on Banach spaces and other topological linear spaces
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