Admissible vectors and Radon-Nikodym theorems

TL;DR

This paper explores the mathematical structure of admissible vectors in group representations, linking them to weights on von Neumann algebras and employing advanced tools like noncommutative Fourier transforms and Radon-Nikodym theorems.

Contribution

It demonstrates that admissible vectors can be characterized as weights with central support on the group von Neumann algebra, using noncommutative analysis techniques.

Findings

Admissible vectors correspond to weights with central support on the group von Neumann algebra.

The analysis employs spatial and cocycle derivatives, and noncommutative $L^p$-Fourier transforms.

Square integrability restricts weights to the predual of the algebra, expressible via a bounded element.

Abstract

Admissible vectors lead to frames or coherent states under the action of a group by means of square integrable representations. This work shows that admissible vectors can be seen as weights with central support on the (left) group von Neumann algebra. The analysis involves spatial and cocycle derivatives, noncommutative $L^p$-Fourier transforms and Radon-Nikodym theorems. Square integrability confine the weights in the predual of the algebra and everything may be written in terms of a (right selfdual) bounded element.