On Unitary Representations Of Groups And Related C~*-Algebras

Unitary representation of group is an important research subject of algebra, geome-try, functional analysis and other branches of mathematics. Recently, research on interdis-ciplinary theory and application of group and C*-algebra has become more and more ac-tive. Starting with unitary representation of group, this dissertation studies almost unitaryrepresentation of Property (T) group, constrained unitary representation of free group F₂and corresponding group C*-algebras, applications of almost convergence to C*-algebraand discrete topological semi-group, etc. The main results obtained are as follows:1. ZË™uk's sufficient condition for Property (T) is generalized to the case of almostunitary representation. This makes the study on the spectrum of averaging operatorÏ€(χ)of Property (T) group with almost unitary representationμpossibly independent fromasymptotic unitary representation, hence extends our research range to non-AGA Property(T) groups.2. The concept ofμ-constrained unitary representation is defined for free group F₂,and we show its existence, abundance and dependence on parameterμ. Correspondinggroup C*-algebraμμis induced byμ-constrained unitary representations. By provingthat (Aμ,I,A/B) is a continuous bundle of C*-algebras, it is shown that A_μs possesscertain continuity with respect toμ. We calculate theμ-groups of A0, by identifyingA₀ with some amalgamated free product of C*-algebras and applying Cuntz'sμ-theoryexact sequences about amalgamated free product. Based on the fact thatμ-groups ofA₀ and A₄ are the same, by constructing homotopy between universal representation andtrivial representation of F₂, we calculate the K-groups of Aμfrom homotopy invarianceof functor K_i.3. Both theory and application of almost convergence are studied respectively. Intheory, finitely additive probability measureμ_ac is defined on N, and it is pointed out thatμ_ac-measurable sequence, i.e., properly distributed sequence, is almost convergent withtheir (unique) Banach limit in the form of formal integral. By introducing the concept ofBanach limit functional, the concept of strong almost convergence is defined for boundedsequences in normed vector space, and it is shown that almost convergence and quasi-almost convergence in some papers are equivalent to our strong almost convergence here, hence we summarize and unify the theory of almost convergence of vector-valued se-quences. In application, we extract a unital commutative C*-algebra–M_T from the spacel^∞_ac(N,C) of almost convergent sequences. By Gelfand transform, it is proved that themaximal ideal space t_N of M_T is a compactification of N containingβN as a closedsubset, while M_T provides an example of non-C*-re?exive C*-algebra for Hilbert C*-module theory. Finally, we assert that applications of almost convergence above could begeneralized to countably infinite left amenable cancellative semi-groups.