Extension and separation properties of positive definite functions on locally compact groups

Abstract

Abstract. Continuing earlier work, we investigate two related aspects of the set P (G) of continuous positive definite functions on a locally compact group G. The first one is the problem of when, for a closed subgroup H of G, every function in P (H) extends to some function in P (G). The second one is the question whether elements in G \ H can be separated from H by functions in P (G) which are identically one on H.

The mathematical work of Anthony To-Ming Lau

“…Geometry of continuous positive definite functions on groups, and group von Neumann algebras. In the paper [17], the authors study Hahn-Banach and separation properties for positive definite functions on groups. Let G be a locally compact group, and let H be a closed subgroup of G. Consider the property that the continuous positive definite functions on G which are identically 1 on H separate points in G \ H from points in H. Forrest [9] shows that if G is a [SIN]-group, then G has this separation property for every closed subgroup.…”

Positive extensions of Schur multipliers

“…The special case where E is a closed subgroup has attracted considerable attention previously (see e.g. [14]). Closely related problems about extension of Herz-Schur multipliers were recently considered in [6].…”

On spherical unitary representations of groups of spheromorphisms of Bruhat--Tits trees

“…A detailed investigation of such phenomena for Lie groups were done by Moore [35] and Wang [58], for p-adic groups by Wang [58]- [59]. Kaniuth, Lau [21] and Losert [32] discussed stabilizers of vectors in unitary representations of general locally compact groups 7 .…”