On T-type Properties For C*-algebras And Banach Algebras

The notion of property T is very useful in the study of topological groups. It was imported to the C*-algebras by Bekka in2005. Since non-unital C*-algebras do not have the direct analogue of such T-property, we define and study a new property named property T**for C*-algebras. In addition, we generalize the notion of property T to unital Banach*-algebras and more generally, to unital Banach algebras.This thesis is organized as follows.The background and development of property T are introduced in Chapter1.Chapter2is devoted to some definitions and facts that we need about von Neumann algebras, group C*-algebras, tensor products of C*-algebras and property T for groups as well as for C*-algebras.In Chapter3, we define and study property T’ for von Neumann algebras as well as property T**for (possibly non-unital) C*-algebras. Besides giving some equivalent formulations with respect to these two properties, we obtain several results of property T**parallel to those of property T for unital C*-algebras. In addition, it is proved that a discrete group Γ has property T if and only if the group C*-algebra C*(T)(or equivalently, the reduced group C*-algebra Cr*(T)) has property T**. We also show that the compact operators K(l2) has property T**but c0does not have property T**In Chapter4, we define property T*for unital Banach*-algebras as well as prop-erty T for unital Banach algebras. It is showed that a unital Banach*-algebra has property T*if and only if its C*-envelop has property T. Moreover, a countable dis-crete group Γ has property T if and only if l1(Γ) has property T as a Banach algebra, which is also equivalent to l1(Γ) has property T*as a Banach*-algebra.