Limits of uniformly infinitesimal families of projective representations of locally compact groups

Abstract

The problem of finding limit theorems for uniformly infinitesimal families of projective or ordinary representations of locally compact groups arose in two different contexts: Gangolli [3] investigated infinitely divisible positive definite functions on certain locally compact groups in connection with the definition of generalized Brownian stochastic processes. The second approach arose from a problem in quantum field theory. Araki [1] and Streater [10] investigated factorisable and infinitely divisible cyclic representations of locally compact groups. Gangolli, Araki and Streater all pointed out the importance of the so-called conditionally positive definite functions, which arise as logarithms of the infinitely divisible and factorisable positive definite functions in all cases, which had been investigated. The general Levy-Khinchin formula for conditionally positive definite and conditionally s-positive definite functions (which arise in the case of infinitely divisible multiplier representations) was given in [7]. The aim of this paper is to show, that every possible limit of a uniformly infinitesimal family of projective or ordinary representations of a locally compact second countable group is described by the Levy-Khinchin formula: It is proved, that every such limit possesses an additive multiplier s and a conditionally s-positive definite function as "logarithm" and is therefore described by the Levy-Khinchin formula. Further it is shown, that if the limit is an ordinary representation, then this logarithm is given by a conditionally positive definite function. In the last chapter it is pointed out that our definition of uniformly infinitesimal families of cyclic multiplier representations is wide enough to include (in a certain sense) all infinitely divisible ordinary as well as projective representations. The problem of finding these logarithms has been solved earlier in various special cases by Gangolli [3], Parthasarthy [4 and 6], Parthasarathy and the author [7 and 8], and the author in [9]. A central limit theorem for a special group was proved by Cushen and Hudson in [2].