Minimal projections, integrable representations and property (T) (Q793161)

A projector p in a \(C^*\)-algebra A is minimal if \(pAp={\mathbb{C}}p\). Using properties of minimal projectors first given by \textit{B. A. Barnes} [Edinb. Math. Soc., II. Ser. 23, 229-238 (1980; Zbl 0449.43006)] we show that, if A is separable unital, there is a one-to-one correspondence between classes of unitary equivalence of minimal projectors in A and open points in the dual \(\hat A\) of A. This is used to study groups having property (T), i.e. groups for which the trivial one-dimensional representation \(1_ G\) is isolated in \(\hat G\). We characterize property (T) by the existence in the group \(C^*\)-algebra \(C^*(G)\) of a central minimal projector \(p_ G\) such that \(1_ G(p_ G)=1\); if G is discrete, \(p_ G\) appears as a spectral projector of \(h+h^*\), where h is the characteristic function of some finite subset of G. We use all this to show that the presence in the group G of a closed normal subgroup H having property (T) forces a direct sum decomposition \(C^*(G)=I\oplus C^*(G/H).\) In particular, G has property (T) if both H and G/H have it. Finally, we give short proofs of two results of \textit{P. S. Wang} [Math. Ann. 218, 19-34 (1975; Zbl 0332.22009)]: let G be a second countable group, and let H be a closed subgroup having property (T); if H is either cocompact or of finite covolume, then G itself has property (T).