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Symbol calculus of square-integrable operator-valued maps. (English) Zbl 1372.46052
The purpose of this paper is to develop an abstract framework to study quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representation. The key concept is that of square-integrable operator-valued maps, generalizing the notion of matrix coefficients of a square-integrable group representation. If \((\Sigma,\mu)\) is a Borel space and \(\varpi\) is a weakly measurable map from \(\Sigma\) to the bounded linear operators on some Hilbert space \(\mathcal{H}\), then \(\varpi\) is said square-integrable if \[ \int_\Sigma\left\langle\varpi(s)\xi_1,\eta_1\right\rangle\left\langle\eta_2,\varpi(s)\xi_2\right\rangle\,d\mu(s)=\left\langle\xi_1,\xi_2\right\rangle\left\langle\eta_1,\eta_2\right\rangle \] holds for all \(\xi_1,\xi_2, \eta_1,\eta_2\) in \(\mathcal{H}\). A symbol calculus can be associated with such a map: functions of the form \(s\mapsto\left\langle\varpi(s)\xi,\eta\right\rangle\) generate a closed subspace \(\mathcal{B}_2(\Sigma)\) of \(L^2(\Sigma)\). It is, in fact, a complete Hilbert algebra which maps to Hilbert-Schmidt operators on \(\mathcal{H}\) via integration: \[ \Pi:f\longmapsto\int_\Sigma f(s)\varpi(s)^*\,d\mu(s). \]
The authors show that the class of square-integrable families of operators is closed under compressions and tensor products and possesses certain irreducibility properties. They proceed to extend the map \(\Pi\) between a subspace of \(L^2(\Sigma)\) and the entire Hilbert-Schmidt ideal.
An important result proved in the paper is the fact that infinite tensor products of square-integrable families of operators are always approximately square-integrable. This is particularly relevant in relation to the representation theory of canonical commutation relations with infinitely many degrees of freedom.
Other topics approached within this framework include the Berezin-Toeplitz quantization, the magnetic Weyl calculus, the metaplectic representation of locally compact abelian groups, representations of infinite-dimensional Lie groups associated with finite-dimensional coadjoint orbits, and the representation theory of nilpotent Lie groups.

MSC:
46L65 Quantizations, deformations for selfadjoint operator algebras
22E66 Analysis on and representations of infinite-dimensional Lie groups
35S05 Pseudodifferential operators as generalizations of partial differential operators
46H30 Functional calculus in topological algebras
46K15 Hilbert algebras
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