Transition operator characterizations of compact and maximally almost periodic locally compact groups.

*(English)*Zbl 0583.47038Some 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.

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 |