Connes-amenability of Fourier–Stieltjes algebras

Abstract

Let G be a locally compact group, and let B(G) denote its Fourier-Stieltjes algebra. We show that B(G) is Connes-amenable if and only if G is almost abelian.

Character Connes amenability of dual Banach algebras

“…Recently, Runde and Uygul in [24] showed that B(G) is Connes amenable if and only if G is almost abelian, i.e., it has an abelian subgroup of finite index. Character amenability of B(G) is also studied in [9], Section 4.…”

A twisted inclusion between tensor products of operator spaces

“…A core result in the theory of operator spaces is the following observation: the operation on K(ℓ 2 ) given by matrix transpose, a → a ⊤ , fails to be completely bounded, even though it is an isometric involution of Banach spaces. This fact serves to explain certain phenomena in non-commutative harmonic analysis, and can be exploited to prove structural results about Fourier and Fourier-Stieltjes algebras of locally compact groups: see, for instance, [4,12].…”

Operator Connes-amenability of completely bounded multiplier Banach algebras

“…Examples of dual Banach algebras (besides von Neumann algebras) include the measure algebra M (G) and the Fourier-Stieltjes algebra B(G) of a locally compact group G. Runde [17] showed that a locally compact group G is amenable if and only if its measure algebra M (G) is Connes-amenable, see also [18]. On the other hand, the Fourier-Stieltjes algebra B(G) is operator Connes-amenable if and only if G is almost abelian [21]. The first and third authors in [5,6] investigated two possible setups in which one could guarantee that the multiplier algebra M(B) of a Banach algebra B is a dual Banach algebra, and found conditions for the equivalence of amenability of B with Connes-amenability of M(B).…”