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Connes-amenability of Fourier-Stieltjes algebras. (English) Zbl 1335.46041

Amenability for Banach algebras was introduced by B. E. Johnson in [Bull. Soc. Math. Fr. 100, 73–96 (1972; Zbl 0234.46066)] and has since grown to be a concept of great importance. A variant of this, now called Connes amenability, appeared in the profound work [Ann. Math. (2) 104, 73–115 (1976; Zbl 0343.46042)] of A. Connes on von Neumann algebras.
A Banach algebra \(A\) which has a predual \(A_{\ast}\) such that multiplication on \(A\) is separately \(\sigma(A,A_{\ast})\)-continuous is called a dual Banach algebra. This was formally defined by the first author in 2001, but the concept itself goes back to the 1972 paper of Johnson-Kadison-Ringrose [B. E. Johnson et al., Bull. Soc. Math. Fr. 100, 73–96 (1972; Zbl 0234.46066)]. A rich theory has been developed since then for this ample class of Banach algebras (which includes several classes of importance, von Neumann algebras, measure algebras and Fourier-Stieljes algebras of locally compact groups, algebras of operators on reflexive Banach spaces etc.) For this class, it has been found that Connes amenability is the more suited one. For example, it is known that the Connes amenability of the measure algebra \(M(G)\) of a locally compact group \(G\) is equivalent to the amenability of \(G\), whereas \(M(G)\) is amenable if and only if \(G\) is a discrete amenable group.
The first author proved in [V. Runde, Math. Scand. 95, No. 1, 124–144 (2004; Zbl 1087.46035)] that the Fourier-Stieltjes algebra \(B(G)\) of a locally compact group \(G\) is Connes amenable if \(G\) is almost abelian, i.e., \(G\) has an abelian subgroup of finite index. He proved the converse in some special cases and conjectured that it holds in full generality. The authors settle this conjecture in this paper by showing that the following are equivalent:
(i) \(B(G)\) is Connes amenable; (ii) \(B(G)\) has a normal virtual diagonal; (iii) \(G\) has an abelian subgroup of finite index.
It may be remarked that the concept of a virtual diagonal was introduced by Johnson who showed that a Banach algebra is amenable if and only if it has a virtual diagonal. A normal virtual diagonal is a variant, introduced by E. G. Effros [J. Funct. Anal. 78, No. 1, 137–153 (1988; Zbl 0655.46053)] for von Neumann algebras.

MSC:

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46H20 Structure, classification of topological algebras
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
22D05 General properties and structure of locally compact groups
46J05 General theory of commutative topological algebras
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References:

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