Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T08:03:47.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  17 December 2020

Zhang Lunchuan
Zhang Lunchuan*
Affiliation:
School of Mathematics, Renmin University of China, Beijing100086, P. R. China ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

Keywords

Quantum Markov semigroupDirichlet formhypercontractivitylogarithmic Sobolev inequality

MSC classification

Primary: 46L53: Noncommutative probability and statistics
Secondary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 1 , February 2021 , pp. 59 - 71
Check for updates
Copyright
Copyright © The Author(s) 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albeverio, S. and Høegh-Krohn, R., Dirichlet forms and Markov semigroups on C*-algebras, Comm. Math. Phys. 77 (1975), 91–102.Google Scholar
Ben-Aroya, A., Regev, O. and de Wolf, R., A hypercontractive inequality for matrix-valued functions with applications to quantum computing and LDC, IEEE Symp. Found. Comput. Sci. (FOCS) 49 (2008), 477–486.Google Scholar
Bialynicki-Birula, I. and Mycielski, J., Uncertainty relations for information entropy in wave mechanics, Comm. Math. Phys. 44 (1975), 129–132.Google Scholar
Biane, P., Free hypercontractivity, Comm. Math. Phys. 184 (1997), 457–474.CrossRefGoogle Scholar
Boz̀ejko, M., Deformed Fock spaces, Hecke operators and monotone Fock space of Muraki, Demonstratio Math. XLV (2012), 129–154.Google Scholar
Boz̀ejko, M. and Speicher, R., An example of a generalized Brownian motion, Comm. Math. Phys. 137 (1991), 519–531.CrossRefGoogle Scholar
Boz̀ejko, M. and Speicher, R., Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994), 97–120.Google Scholar
Boz̀ejko, M., Kümmerer, B. and Speicher, R., q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), 399–413.Google Scholar
Carbone, R. and Sasso, E., Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup, J. Funct. Anal. Probab. Theory Relat. Fields 140 (2008), 505–522.Google Scholar
Carlen, E. A. and Lieb, E. H., Optimal hypercontractivity for Fermi fields and related noncommutative integration inequalities, Comm. Math. Phys. 155 (1993), 27–46.CrossRefGoogle Scholar
Cipriani, F., Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras, J. Funct. Anal. 147 (1997), 259–300.CrossRefGoogle Scholar
Davies, E. B. and Lindsay, J. M., Non-commutative symmetric Markov semigroups, Math. Zeit. 210 (1992), 379–411.Google Scholar
Franz, U., Guixiang, H., Ulrich, F. M. and Haonan, Z., Hypercontractivity of heat semigroups on free quantum groups, J. Operator Theory. 77(1) (2017), 61–76.CrossRefGoogle Scholar
Fuglede, B. and Kadison, R. V., Determinant theory in finite factors, Ann. Math. 55(3) (1952), 520–530.Google Scholar
Gross, L., Hypercontractivity and logarithmic Sobolev inequalities for the Clifford Dirichlet form, Duke Math. J. 43 (1975), 383–396.Google Scholar
Gross, L., Logarithmic Sobolev inequalities, Am. J. Math. 97 (1975), 1061–1083.Google Scholar
Junge, M., Palazuelos, C., Parcbt, J., Perrin, M. and Ricard, E., Hypercontractivity for free products, Ann. Sci. Ecole. Norm. Sup. 48(4) (2015), 861–889.CrossRefGoogle Scholar
Kosaki, H., Applications of uniform convexity of noncommutative L p spaces, Trans. AMS 283(1) (1984), 265–282.Google Scholar
Królak, I., Wick product for commutation relations connected with Yang-Baxter operators and new constructions of factors, Comm. Math. Phys. 210 (2000), 685–700.Google Scholar
Królak, I., Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations, Math. Zeit 25 (2005), 915–937.CrossRefGoogle Scholar
Królak, I., Optimal holomorphic hypercontractivity for CAR algebras, Bull. Polish Acad. Sci. 58 (2010), 79–90.CrossRefGoogle Scholar
Lunchuan, Z. and Maozheng, G., The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert C*-module l 2(A), Sci. China Math. 57(2) (2014), 377–387.Google Scholar
Lust-Piquard, F., Riesz transforms on deformed Fock spaces, Comm. Math. Phys. 205 (1999), 519–549.Google Scholar
Mingchu, G., Free Ornstein-Uhlenbeck processes, J. Math. Anal. Appl. 322(1) (2006), 177–192.Google Scholar
Mingchu, G., Free Markov processes and stochastic differential equations in von Neumann algebras, Illinois J. Math. 52 (2008), 153–180.Google Scholar
Nelson, E., A quartic interaction in two dimensions, in Mathematical theory of elementary particles (eds. Goodman, R. and Segal, I.E.), pp. 67–73 (M.I.T. Press, Cambridge, Mass, 1965).Google Scholar
Nielsen, M. and Chuang, L., Quantum computation and quantum information (Cambridge, Cambridge University Press, 2000).Google Scholar
Olkiewicz, R. and Zegarlinski, B., Hypercontractivity in noncommutative L p spaces, J. Funct. Anal. 161(1) (1999), 246–285.CrossRefGoogle Scholar
Ricard, E. and QuanHua, X., A noncommutative martingale convexity inequality, Ann. Probability 44(2) (2016), 867–882.Google Scholar
Villani, C., Topics in optimal transportation, 300, pp. 97–120 (Springer, 2003) (1994).Google Scholar