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.

Keywords: Position and momentum operators, quadratures operators, $C^*$-algebras

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