Finite dimensional approximations and deformations of group C*-algebras

Quasidiagonality is a finite-dimensional approximation property of a C*-algebra which indicates that it has matricial approximations that capture the structure of the C*-algebra. We investigate when C*-algebras associated to discrete groups have such a property with particular emphasis on finding obstructions. In particular, we point out that groups with Kazhdan\u27s Property (T) and only finitely many unitary equivalence classes of finite dimensional representations do not produce quasidiagonal C*-algebras. We then observe and note interactions with Property (T) and other approximation properties. Property (QH) is a related but stronger approximation property with deep connections to E-Theory and KK-Theory. In the case of groups, Property (QH) represents the property that the group not only has structure capturing matricial models but also that these models may be deformed to the trivial representation. In this sense, Property (QH) may then be considered a type of finite-dimensional deformation property. In joint work with Marius Dadarlat and Ulrich Pennig, we show the class of groups with Property (QH) is closed under wreath products, thus producing a new class of highly non-trivial groups with Property (QH) far from those currently known