\(C^*\)-algebras associated with groups with Kazhdan's property T (Q1185452)

A countable, discrete group \(G\) has Kazhdan's property \(T\) if there is a finite subset \(g_ 1,\dots,g_ n\) and a positive \(\varepsilon\) such that, whenever \(\pi\) is a unitary representation of \(G\) on a Hilbert space \(H\) which contains a unit vector \(\xi\) for which \(\|\pi(g_ i)\xi-\xi\|<\varepsilon\) (\(i=1,2,\dots\)) there is a non-zero vector in \(H\), invariant under \(\pi(G)\). Such groups were used by Connes to construct a type \(II_ 1\) factor with countable fundamental group. The author employs them to construct further examples, in particular a quasi- diagonalizable \(C^*\)-algebra \(A\) of operators on a Hilbert space with a quotient \(B\) with respect to the compact operators in \(A\) which is not quasi-diagonalizable. Further, \(B\) is such that the space \(\text{Ext}(B)\) is not a group.