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Markov dilations of semigroups of Fourier multipliers

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Abstract

We describe a Markov dilation for any weak* continuous semigroup \((T_t)_t \geqslant 0\) of selfadjoint unital completely positive Fourier multipliers acting on the group von Neumann algebra \(\mathrm {VN}(G)\) of a locally compact group G.

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Notes

  1. That means that for any \(f \in \mathrm {L}^\infty (\Omega )\) we have \(\int _{\Omega } \Gamma ^\infty (u)f \mathop {}\mathopen {}\mathrm {d}\mu =\int _{\Omega } f \mathop {}\mathopen {}\mathrm {d}\mu \).

  2. The existence of a proof of Lemma 2.1 without (2.4) is unclear.

  3. That means that \(\mathrm {B}(\mathrm {L}^\infty (\Omega ))\) is equipped with the point weak* topology.

  4. In the book [39], the author considers weak* continuous functions, it is problematic since the product of M is not weak* continuous even on bounded sets by [31, Exercise 5.7.9] (indeed this latter fact is equivalent to the weak continuity of the product on bounded sets).

  5. The function \(u :G \rightarrow \mathrm {U}(M)\) is a \(\alpha \)-1-cocycle.

References

  1. Anantharaman-Delaroche, C.: Action moyennable d’un groupe localement compact sur une algèbre de von Neumann. (French). Math. Scand. 45(2), 289–304 (1979)

  2. Anantharaman-Delaroche, C.: On ergodic theorems for free group actions on noncommutative spaces. Probab. Theory Related Fields 135(4), 520–546 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arhancet, C.: On Matsaev’s conjecture for contractions on noncommutative \(L^p\)-spaces. J. Oper. Theory 69(2), 387–421 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arhancet, C.: Dilations of semigroups on von Neumann algebras and noncommutative \(\rm L^p\)-spaces. J. Funct. Anal. 276(7), 2279–2314 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arhancet, C.: Dilations of Markovian semigroups of Fourier multipliers on locally compact groups. Proc. Am. Math. Soc. 148(6), 2551–2563 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arhancet, C.: Dilations of markovian semigroups of measurable Schur multipliers. Preprint, arXiv:1910.14434

  7. Arhancet, C., Fackler, S., Le Merdy, C.: Isometric dilations and \(H^\infty \) calculus for bounded analytic semigroups and Ritt operators. Trans. Am. Math. Soc. 369, 6899–6933 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arhancet, C., Le Merdy, C.: Dilation of Ritt operators on \(L^{p}\)-spaces. Israel J. Math. 201(1), 373–414 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Arhancet, C., Kriegler, C.: Projections, multipliers and decomposable maps on noncommutative \(L^p\)-spaces. Preprint, arXiv:1707.05591. To appear in Mémoires de la Société Mathématique de France

  10. Arhancet, C., Kriegler, C.: Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers. Lecture Notes in Mathematics, vol. 2304. Springer, Cham (2022)

  11. Bekka, B., de la Harpe, P., Valette, A.: Kazhdan’s Property (T). New Mathematical Monographs, 11. Cambridge University Press, Cambridge (2008)

  12. Bourbaki, N.: Integration. I. Chapters 1–6. Translated from the 1959, 1965 and 1967 French originals by Sterling K. Berberian. Elements of Mathematics. Springer, Berlin (2004)

  13. Brannan, M., Gao, L., Junge, M.: Complete Logarithmic Sobolev Inequalities via Ricci Curvature Bounded Below II. Preprint, arXiv:2008.12038

  14. Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics. 1. \(C^*\)- and \(W^*\)-Algebras, Symmetry Groups, Decomposition of States. Second edition. Texts and Monographs in Physics. Springer, New York (1987)

  15. Brown, N.P., Ozawa, N.: \(C^*\)-Algebras and Finite-dimensional Approximations. Graduate Studies in Mathematics, 88. American Mathematical Society, Providence, RI (2008)

  16. Caspers, M., Junge, M., Sukochev, F., Zanin, D.: BMO-estimates for non-commutative vector valued lipschitz functions. J. Funct. Anal. 278(3), 108317, 39 pp (2020)

  17. Caspers, M.: Harmonic analysis and BMO-spaces of free Araki-Woods factors. Studia Math. 246(1), 71–107 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fendler, G.: Dilations of one parameter semigroups of positive contractions on \(L^p\) spaces. Canad. J. Math. 49(4), 736–748 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ferguson, T., Mei, T., Simanek, B.: \(H^\infty \)-calculus for semigroup generators on BMO. Adv. Math. 347, 408–441 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Haagerup, U.: An example of a nonnuclear C*-Algebra, which has the metric approximation property. Invent. Math. 50(3), 279–293 (1978/79)

  21. Haagerup, U.: On the dual weights for crossed products of von Neumann algebras. I. Removing separability conditions. Math. Scand. 43(1), 99–118 (1978/79)

  22. Haagerup, U., Junge, M., Xu, Q.: A reduction method for noncommutative \(L_p\)-spaces and applications. Trans. Am. Math. Soc. 362(4), 2125–2165 (2010)

    Article  MATH  Google Scholar 

  23. Haagerup, U., Musat, M.: Factorization and dilation problems for completely positive maps on von Neumann algebras. Commun. Math. Phys. 303(2), 555–594 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hytönen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis in Banach Spaces, Volume I: Martingales and Littlewood-Paley theory. Springer, Berlin (2016)

  25. Hytönen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis on Banach Spaces, Volume II: Probabilistic Methods and Operator Theory. Springer, Berlin (2018)

  26. Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, 129. Cambridge University Press, Cambridge (1997)

  27. Jiao, Y., Wang, M.: Noncommutative harmonic analysis on semigroups. Indiana Univ. Math. J. 66(2), 401–417 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Junge, M., Mei, T.: Noncommutative Riesz transforms-a probabilistic approach. Am. J. Math. 132(3), 611–680 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Junge, M., Mei, T.: BMO spaces associated with semigroups of operators. Math. Ann. 352(3), 691–743 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Junge, M., Ricard, E., Shlyakhtenko, D.: In preparation

  31. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras. Vol. I. Elementary theory. Reprint of the 1983 original. Graduate Studies in Mathematics, 15. American Mathematical Society, Providence, RI (1997)

  32. Kaniuth, E., Lau, A. T.-M.: Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups. Mathematical Surveys and Monographs, 231. American Mathematical Society, Providence, RI (2018)

  33. Neerven, J.V.: Stochastic Evolution Equations. ISEM Lecture Notes 2007/08

  34. Neerven, J.V., Veraar, M., Weis, L.: Stochastic Integration in Banach Spaces: A Survey. Stochastic Analysis: A Series of Lectures, pp. 297–332 (2015)

  35. Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  36. Osborne, M.S.: Locally Convex Spaces. Graduate Texts in Mathematics, 269. Springer, Cham (2014)

  37. Pedersen, G.K.: \(C^{\ast } \)-algebras and their Automorphism Groups. London Mathematical Society Monographs, 14. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York (1979)

  38. Ricard, É.: A Markov dilation for self-adjoint Schur multipliers. Proc. Am. Math. Soc. 136(12), 4365–4372 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Stratila, S.: Modular Theory in Operator Algebras. Translated from the Romanian by the author. Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press, Tunbridge Wells (1981)

  40. Takesaki, M.: Theory of Operator Algebras. II. Encyclopaedia of Mathematical Sciences, 125. Operator Algebras and Non-commutative Geometry, 6. Springer, Berlin (2003)

  41. Takesaki, M.: Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math. 131, 249–310 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wirth, M.: The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups. Preprint, arXiv:2207.09247

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Acknowledgements

The author acknowledges support by the grant ANR-18-CE40-0021 (project HASCON) of the French National Research Agency ANR. The author is grateful to the referee for some corrections and a simplification of the proof.

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    Cédric Arhancet

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Arhancet, C. Markov dilations of semigroups of Fourier multipliers. Positivity 26, 83 (2022). https://doi.org/10.1007/s11117-022-00950-w

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