Transition operator characterizations of compact and maximally almost periodic locally compact groups. (English) Zbl 0583.47038

Glicksberg, Irving

Math. Scand. 55, 121-140 (1984).

Some results about transition operators (i.e. non-negative operators fixing the constants) on the space C(X) of continuous functions defined over a compact Hausdorff space X, are presented. The main emphasis is given on eigenvectors which characterize those families of transition operators that arise from the action of a compact transformation group. The indicated groups are characterized by a set of translation operators, and also rotations, as transition operators on spheres.
First one considers the case of compact groups. The following properties of the set \({\mathfrak T}\) of transition operators on C(X) are analysed: (a) for each \(T\in {\mathfrak T}\) the probability measures fixed by \(T^*\) have the union of their supports dense in X; (b) there is a set of common \(T^ 1\)-eigenvectors (i.e. with eigenvalues of modulus 1) of the elements of \({\mathfrak T}\) which separate the points of X; (c) \({\mathfrak T}\) fixes only the constants. The Abelian and non-Abelian versions are investigated.
Secondly, the case of locally compact groups is described. A new result about almost periodic semigroups is obtained: if \({\mathfrak G}\) is a uniformly bounded semigroup of operators on a Banach space \(B\) and the span of the \({\mathfrak G}\)-unitary subspaces is dense in B, then its strong operator closure \({\mathfrak G}^-\) is a compact group of invertibles in the strong operator topology.

Reviewer: G.Zet

MSC:

47B38	Linear operators on function spaces (general)
47D03	Groups and semigroups of linear operators
54C40	Algebraic properties of function spaces in general topology
22D05	General properties and structure of locally compact groups

Keywords:

transition operators; eigenvectors; action of a compact transformation group; translation operators; rotations; locally compact groups; almost periodic semigroups; compact group of invertibles in the strong operator topology