Journal of Operator Theory

Volume 52, Issue 2, Fall 2004 pp. 259-266.

$C^*$ -Algebras of quadratures

Authors: Henri Comman (1) and Franco Fagnola (2)
Author institution: (1) Department of Mathematics, University of Santiago de Chile, Bernardo O'Higgins 3363 Santiago, Chile
(2) Univesita degli Studi di Genova, Dipartimento di Matematica, Via Dodecaneso 35, I - 16146 Genova, Italy

Summary: Quadrature operators are the

$q_{\theta}= (\e^{-{\rm i}\theta}a +\e^{{\rm i}\theta}a^*)/{\sqrt{2}}$ where

$a$ and

$a^*$ are the annihilation and creation operators on

$L^2(\mathbb{R})$ . The structure of the

$C^*$ -algebra generated by operators

$f(q_\theta)$ for

$f$ continuous function vanishing at infinity and

$\theta$ in any subset

$\Theta$ of

$]-\pi,\pi[$ with

$\hbox{Card}(\Theta)\geq 2$ is studied. It is shown that it contains all compact operators and it is a

$C^*$ -algebra of type I. Its atomic representation and the structure of its spectrum is explicitely given. A trace formula for the operators

$f(q_{\theta_1})g(q_{\theta_2})$ is proved.

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