M ay 2 00 4 A Connes - amenable , dual Banach algebra need not have a normal , virtual diagonal

Let G be a locally compact group, and let WAP(G) denote the space of weakly almost periodic functions on G. We show that, if G is a [SIN]-group, but not compact, then the dual Banach algebra WAP(G)∗ does not have a normal, virtual diagonal. Consequently, whenever G is an amenable, non-compact [SIN]-group, WAP(G)∗ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual diagonal. On the other hand, there are amenable, non-compact, locally compact groups G such that WAP(G)∗ does have a normal, virtual diagonal.