IRREDUCIBLE DECOMPOSITIONS OF UNITARY REPRESENTATIONS AND EXTREMAL DECOMPOSITIONS OF POSITIVE-DEFINITE FUNCTIONS ON GROUPS

摘要

This paper concerns with two kind of decompositions of unitary representations or positive-definite functions on groups. The first one is a specific example of irredeucible decompositions of unitary representations of infinite-dimensional groups which was already obtained in [7], and which is regarded as an infinite-version descibed in [11]. In the second part, we consider positive-definite functions (PDF) 0 on the usual locally compact groups. Φ has an extremal decomposition through an irreducible decomposition of the unitary representation corresponding to Φ (by the GNS construction method). However there are other ways to get extremal decompositions, for example via the Chocqet theorem. So it is interesting to find conditions which disdinguish the natural particular one from other decompositions. We describe a necessary and sufficient condition for the above problem as well as an interesting negative example related to [4, 10].