Admissible vectors and traces on the commuting algebra

TL;DR

This paper explores the relationship between admissible vectors, traces, and the structure of the commuting algebra in group representations, providing new criteria and proofs for admissibility and biorthogonality in harmonic analysis.

Contribution

It introduces a trace-based criterion for admissibility, generalizes Wexler-Raz relations, and offers new proofs for admissibility in type I representations.

Findings

Existence of a unique finite trace characterizes admissible vectors.

Admissibility criteria are connected to the central decomposition of the regular representation.

Generalization of Wexler-Raz biorthogonality relations for Weyl-Heisenberg frames.

Abstract

Given a representation of a unimodular locally compact group, we discuss criteria for associated coherent state expansions in terms of the commuting algebra. It turns out that for those representations that admit such expansions there exists a unique finite trace on the commuting algebra such that the admissible vectors are precisely the tracial vectors for that trace. This observation is immediate from the definition of the group Hilbert algebra and its associated trace. The trace criterion allows to discuss admissibility in terms of the central decomposition of the regular representation. In particular, we present a new proof of the admissibility criteria derived for the type I case. In addition we derive admissibility criteria which generalize the Wexler-Raz biorthogonality relations characterizing dual windows for Weyl-Heisenberg frames.