Exotic group C*-algebras in noncommutative duality

Abstract

We show that for a locally compact group G there is a one-to-one correspondence between G-invariant weak*-closed subspaces E of the Fourier-Stieltjes algebra B(G) containing B_r(G) and quotients C*_E(G) of C*(G) which are intermediate between C*(G) and the reduced group algebra C*_r(G). We show that the canonical comultiplication on C*(G) descends to a coaction or a comultiplication on C*_E(G) if and only if E is an ideal or subalgebra, respectively. When \alpha is an action of G on a C*-algebra B, we define "E-crossed products" B\rtimes_{\alpha,E} G lying between the full crossed product and the reduced one, and we conjecture that these "intermediate crossed products" satisfy an "exotic" version of crossed-product duality involving C*_E(G).