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Toeplitz operators and quantum mechanics. (English) Zbl 0626.47031

Toeplitz operators on the Segal-Bargmann spaces of Gaussian measure square-integrable entire functions on complex n-space \({\mathbb{C}}^ n\) are studied. The \(C^*\)-algebra generated by the Weyl form of the canonical commutation relations consists precisely of the uniform limits of almost- periodic Toeplitz operators. The question of “which Toeplitz operators admit a symbol calculus modulo the compact operators” is raised and sufficient conditions are given for such a calculus. These conditions involve a notion of “slow oscillation at infinity”.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46L05 General theory of \(C^*\)-algebras
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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