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The problem of harmonic analysis on the infinite-dimensional unitary group. (English) Zbl 1036.43002

The problem of noncommutative harmonic analysis is to decompose ”natural” unitary representations of a given group into irreducible ones.

The article is concerned with the problem of harmonic analysis in the case when the group is not compact or locally compact, its dual space has infinite dimension, and the decomposition into irreducibles is defined by a measure with infinite-dimensional support. The infinite-dimensional unitary group $U\left(\infty \right)$ is one of the basic examples of “big” groups whose irreducible representations depend on infinitely many parameters. The purpose of the article is to explain what the harmonic analysis on $U\left(\infty \right)$ consists of.

The author deals with unitary representations of a reasonable class, which are in 1-1 correspondence with the characters (central, positive definite, normalized functions on $U\left(\infty \right)$). The decomposition of any representation of this class is described by a probability measure (called spectral measure) on the space of indecomposable characters. The indecomposable characters were found by Dan Voiculescu in 1976.

The author constructs a family of representations that he considers as “natural” ones. In the situation under consideration, when the group is not locally compact, the conventional definition of a regular representation (or a quasi-regular representation associated with a homogeneous space) is not applicable directly. So, the author had to choose another, more sophisticated, way to produce representations.

Some necessary general theorems concerning the spectral measures with infinite-dimensional support are proved.

This allows to convert the problem of harmonic analysis to an asymptotic problem of the form which is typical for random matrix theory or asymptotic combinatorics. As the author notes, he presents a new model that was not previously examined.

A few auxiliary general results are also proved. In particular, it is proved that the spectral measure of any character of $U\left(\infty \right)$ can be approximated by a sequence of (discrete) spectral measures for the restrictions of the character to the compact unitary groups $U\left(N\right)$. This fact is a starting point for computing spectral measures.

The article contains an extended bibliography that reflects several results obtained in this area.

##### MSC:
 43A65 Representations of groups, semigroups, etc. (abstract harmonic analysis)