On a certain invariant of a locally compact group

On a Certain Invariant of a Locally Compact Group

Group here always means a locally compact Hausdorff group, subgroup means a closed subgroup. Let G be a group, H a subgroup and G/H the locally compact homogeneous space of left cosets x = xH. We denote by $(G) [®(G/H)] the family of all compact subsets of G [G/H], The group G acts on G/H in a natural way. If X C.G and Y QG/H, write XY for the set of all elements xy, x £ I , j £ Y. Now assume t...

Vector-valued Invariant Means on Spaces of Bounded Operators Associated to a Locally Compact Group

The purpose of this paper is to introduce and study the notion of a vector-valued π-invariant mean associated to a unitary representation π of a locally compact groupG on S, a self-adjoint linear subspace containing I of B(Hπ). We obtain, among other results, an extension theorem for π-invariant completely positive maps and π-invariant means which characterizes amenability of G. We also study v...

On Marginal Automorphisms of a Group Fixing the Certain Subgroup

Let W be a variety of groups defined by a set W of laws and G be a finite p-group in W. The automorphism α of a group G is said to bea marginal automorphism (with respect to W), if for all x ∈ G, x−1α(x) ∈ W∗(G), where W∗(G) is the marginal subgroup of G. Let M,N be two normalsubgroups of G. By AutM(G), we mean the subgroup of Aut(G) consistingof all automorphisms which centralize G/M. AutN(G) ...

Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups

We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...

A Note on Semi-groups in a Locally Compact Group