Amenability for dual Banach algebras
By Volker Runde. 2002
127 citations recorded by the source corpus.
We dene a Banach algebra A to be dual if A = (A ) for a closed sub- module A of A. The class of dual Banach algebras includes all W -algebras, but also all algebras M(G) for locally compact groups G, all algebrasL(E) for reexiv e Banach spacesE, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We conrm this impression. We rst show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus nite-dimensional . We then develop two no- tions of amenability|Connes amenability and strong Connes amenability|which take the w -topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra A to the (strong) Connes amenability of A ; as an application, we show that there are reexiv e Banach spaces with the approximation property such that L(E) is not Connes amenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudo-measures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones, which avoids the equivalence of amenability and nuclearity for C -algebras.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/fd7deff297f0d37909e400b3b99479a48cafae76
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In Algebra and Number Theory
In Mathematical Physics
Citations in the corpus
- Multipliers, Self-Induced and Dual Banach Algebras (2010, background): “Following [47, 10], we say that a Banach algebra A which is the dual of a Banach space A∗ is a dual Banach algebra if multiplication in A is separately weak-continuous.”
- Dual Banach algebras: Representations and injectivity (2006, methodology): “This corollary is comparable to [17, Theorem 2.2], as a quasi-uniform tensor norm is tight in the sense of [17], and by [ 30 ], the amenability of A(E) is equivalent to the Connes-amenability of B(E), at least when E is reflexive and has the approximation property.”
- Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule (2003, methodology): “In [18], the author extended the notion of Connes-amenability to the larger class of dual Banach algebras (a Banach algebra is called dual if it is a dual Banach space such that multiplication is separatelyw∗-continuous).”
- Connes‐Amenability And Normal, Virtual Diagonals For Measure Algebras I (2001, background): “Paterson showed that, for an inner amenable group G, the group von Neumann algebra VN(G) = PM 2(G) is Connes-amenable if and only if G is amenable; this is generalized to PMp(G) for arbitrary p ∈ (1,∞) in [22].”