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Hypercontractivity in non-commutative holomorphic spaces. (English) Zbl 1100.46035
Summary: We prove an analog of Janson’s strong hypercontractivity inequality in a class of non-commutative “holomorphic” algebras. Our setting is the \(q\)-Gaussian algebras \(\Gamma_{q}\) associated to the \(q\)-Fock spaces of Bozejko, Kümmerer and Speicher, for \(q \in [-1,1]\). We construct subalgebras \(\mathcal H_q \subset \Gamma_q\), a \(q\)-Segal–Bargmann transform, and prove Janson’s strong hypercontractivity \(L^2(\mathcal H_q) \rightarrow L^r(\mathcal H_q)\) for \(r\) an even integer.

MSC:
46L53 Noncommutative probability and statistics
46L51 Noncommutative measure and integration
46L52 Noncommutative function spaces
46L60 Applications of selfadjoint operator algebras to physics
81S05 Commutation relations and statistics as related to quantum mechanics (general)
60H99 Stochastic analysis
30H05 Spaces of bounded analytic functions of one complex variable
30G30 Other generalizations of analytic functions (including abstract-valued functions)
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