Hypercontractivity in non-commutative holomorphic spaces. (English) Zbl 1100.46035

Kemp, Todd

Commun. Math. Phys. 259, No. 3, 615-637 (2005).

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)

References:

[1]	Biane, P.: Free hypercontractivity. Commun. Math. Phys. 184, 457–474 (1997) · Zbl 0874.46049 · doi:10.1007/s002200050068
[2]	Biane, P.: Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems. J. Funct. Anal. 144, 232–286 (1997) · Zbl 0889.47013 · doi:10.1006/jfan.1996.2990
[3]	Baez, J.C., Segal, I.E., Zhou, Z.: Introduction to algebraic and constructive quantum field theory. Princeton Series in Physics. Princeton, Princeton University Press, 1992 · Zbl 0760.46061
[4]	Bozejko, M., Speicher, R.: An example of a generalized Brownian motion. Commun. Math. Phys. 137, 519–531 (1991) · Zbl 0722.60033 · doi:10.1007/BF02100275
[5]	Bozejko, M., Kümmerer, B., Speicher, R.: q-Gaussian processes: non-commutative and classical aspects. Commun. Math. Phys. 185, 129–154 (1997) · Zbl 0873.60087 · doi:10.1007/s002200050084
[6]	Carlen, E., Lieb, E.: Optimal hypercontractivity for Fermi fields and related non-commutative integration inequalities. Commun. Math. Phys. 155, 26–46 (1993) · Zbl 0796.46054 · doi:10.1007/BF02100048
[7]	Dykema, K.; Nica, A.: On the Fock representation of the q-commutation relations. J. Reine Angew. Math. 440, 201–212 (1993) · Zbl 0767.46038
[8]	Gross, L.: Existence and uniqueness of physical ground states. J. Funct. Anal. 10, 52–109 (1972) · Zbl 0237.47012 · doi:10.1016/0022-1236(72)90057-2
[9]	Gross, L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975) · Zbl 0318.46049 · doi:10.2307/2373688
[10]	Gross, L.: Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form. Duke Math. J. 42, 383–396 (1975) · Zbl 0359.46038 · doi:10.1215/S0012-7094-75-04237-4
[11]	Gross, L.: Hypercontractivity over complex manifolds. Acta. Math. 182, 159–206 (1999) · Zbl 0983.47026 · doi:10.1007/BF02392573
[12]	Gross, L.: Hypercontractivity, logarithmic Sobolev inequalities and applications: a survey of surveys. To appear in Diffusion, quantum theory, and radically elementary mathematics: a celebration of Edward Nelson’s contributions to science. Edited by William G. Faris. Princeton University Press, 2006
[13]	Glimm, J.: Boson fields with nonlinear self-interaction in two dimensions. Commun. Math. Phys. 8, 12–25 (1968) · Zbl 0173.29903 · doi:10.1007/BF01646421
[14]	Janson, S.: On hypercontractivity for multipliers on orthogonal polynomials. Ark. Math. 21, 97–110 (1983) · Zbl 0516.42022 · doi:10.1007/BF02384302
[15]	Janson, S., Peetre, J., Rochberg, R.: Hankel forms and the Fock space. Rev. Mat. Iberoamericana 3(1), 61–138 (1987) · Zbl 0704.47022
[16]	Krolak, I.: Contractivity properties of the Ornsetein-Uhlenbeck semigroup for general commutation realtions. To appear in Mathematische Zeitschrift
[17]	Lindsay, J.M.: Gaussian hypercontractivity revisited. J. Funct. Anal. 92, 313–324 (1990) · Zbl 0713.46050 · doi:10.1016/0022-1236(90)90053-N
[18]	Lindsay, J.M., Meyer, P.A.: Fermionic hypercontractivity. In: Quantum probability VII, Singapore: World Scientific, 1992, pp. 211–220 · Zbl 0797.60090
[19]	Meyer, P.A.: Quantum Probability for Probabilists. Second edition, Lecture Notes in Mathematics, Vol. 1538, Berlin–Heidelberg–New York: Springer, 1995 · Zbl 0877.60079
[20]	Nelson, E.: A quartic interaction in two dimensions. In: Mathematical Theory of Elementary Particles, Cambridge, MA: M.I.T. Press, 1965, pp. 69–73
[21]	Nelson, E.: The free Markov field. J. Funct. Anal. 12, 211–227 (1973) · Zbl 0273.60079 · doi:10.1016/0022-1236(73)90025-6
[22]	Pisier, G., Xu, Q.: Non-commutative Lp-spaces. In: Handbook of the geometry of Banach spaces, Vol. 2, Amsterdam: North-Holland, 2003, pp. 1459–1517 · Zbl 1046.46048
[23]	Ricard, E.: Factoriality of q-Gaussian von Neumann algebras. Commun. Math. Phys. 257, 659–665 (2005) · Zbl 1079.81038 · doi:10.1007/s00220-004-1266-5
[24]	Speicher, R.: A non-commutative central limit theorem. Math. Zeit. 209, 55–66 (1992) · Zbl 0737.60020 · doi:10.1007/BF02570820
[25]	Segal, I.: Tensor algebras over Hilbert spaces, II. Ann. of Math. 63, 160–175 (1956) · Zbl 0073.09403 · doi:10.2307/1969994
[26]	Segal, I.: Construction of non-linear local quantum processes, I. Ann. of Math. 92, 462–481 (1970) · Zbl 0213.40904 · doi:10.2307/1970628
[27]	Śniady, P.: Gaussian random matrix models for q-deformed Gaussian variables. Commun. Math. Phys. 216, 515–537 (2001) · Zbl 1021.81025 · doi:10.1007/s002200000345
[28]	Voiculescu, D.V.: Symmetries of some reduced free product C* algebras. In: Operator Algebras and their Connection with Topology and Ergodic Theory, Lecture Notes in Mathematics, Vol. 1132, Berlin-Heidelberg-New York: Springer, 1985, pp. 566–588