[Paper Review] Admissible vectors and traces on the commuting algebra

Hartmut Fuehr|ArXiv.org|Sep 19, 2002

Mathematical Analysis and Transform Methods13 references3 citations

TL;DR

This paper establishes a trace criterion for admissible vectors in unitary representations of unimodular locally compact groups, showing that admissible vectors correspond exactly to tracial vectors for a unique finite trace on the commuting algebra of the regular representation. The key contribution is a generalization of the Wexler-Raz biorthogonality relations in Gabor analysis via the trace on the von Neumann algebra associated with the group Hilbert algebra.

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.

Motivation & Objective

• To develop a general criterion for the existence and characterization of admissible vectors in unitary representations of unimodular locally compact groups.
• To replace irreducible decomposition with the central decomposition of the regular representation for analyzing admissibility.
• To establish a connection between admissible pairs and traces on the commuting von Neumann algebra $VN_r(G)$.
• To generalize the Wexler-Raz biorthogonality relations for Weyl-Heisenberg frames using trace-theoretic methods.
• To provide a new proof of admissibility criteria in the type I case using the trace structure on the commuting algebra.

Proposed method

• Utilizes the group Hilbert algebra and its associated faithful, normal, semifinite trace on the right group von Neumann algebra $VN_r(G)$.
• Defines admissible pairs $(\eta, \psi)$ via the condition $V_\eta^* V_\psi = \text{Id}$ on the representation space $\mathcal{H}_\pi$.
• Establishes that admissible vectors are precisely the tracial vectors for a unique finite trace on $VN_r(G)$, linking admissibility to trace theory.
• Applies the central decomposition of the regular representation $\lambda_G$ to reduce the characterization of admissible pairs to fiber von Neumann algebras.
• Derives orthogonality relations in the form of generalized Wexler-Raz conditions by exploiting the trace structure.
• Applies the framework to the Weyl-Heisenberg group, recovering the standard biorthogonality condition for dual windows in Gabor frames.

Experimental results

Research questions

• RQ1 What is the role of the trace on the commuting algebra in characterizing admissible vectors for unitary group representations?
• RQ2 How can the central decomposition of the regular representation be used to derive admissibility criteria beyond the type I case?
• RQ3 Can the Wexler-Raz biorthogonality relations for Gabor frames be derived from a general trace-theoretic framework?
• RQ4 What is the precise relationship between admissible vectors and tracial vectors in the context of the group Hilbert algebra?
• RQ5 How does the trace criterion simplify or generalize existing admissibility conditions in the literature?

Key findings

• There exists a unique finite trace on the commuting von Neumann algebra $VN_r(G)$ such that the admissible vectors for a representation are exactly the tracial vectors for this trace.
• The admissibility of a pair $(\eta, \psi)$ is equivalent to the condition $V_\eta^* V_\psi = \text{Id}$, which is characterized via the trace on $VN_r(G)$.
• For the Weyl-Heisenberg group, the condition $\langle M_{m/\beta}T_{n/\alpha}\gamma, g \rangle = \alpha\beta \delta_{m,0}\delta_{n,0}$ is derived as a special case of the general trace criterion.
• The paper provides a new proof of the admissibility criterion in the type I case using trace-theoretic decomposition, generalizing earlier results.
• The trace criterion allows the characterization of admissible pairs through orthogonality relations in the dual lattice, generalizing the Wexler-Raz condition.
• The vector $\eta_0 = \sqrt{\alpha} \chi_{[0,\beta)}$ is shown to be admissible for the Weyl-Heisenberg representation, confirming the trace-based criterion.