Exotic group C*-algebras in noncommutative duality

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 α is an action of G on a C*-algebra B, we define E-crossed products B ⋊ α,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).