A character of a finite group G is defined as the trace of a matrix representation of the group G. This notion can be generalized to the class of infinite groups by generalizing relevant properties of the matrix trace. Namely, a function f: G-> C is a "normalized" character if f(1)=1, f(ab) = f(ba) and f is positive-definite. Vershik observed that every measure preserving action of a group G on a probability measure space (X,mu) gives rise to a character on G via the function f(g) = mu(FixedPoints(g)). In this talk we discuss several classes of groups for which the only characters are those that come from measure-preserving actions.
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