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Absolute dilations of UCP self-adjoint Fourier multipliers: the nonunimodular case

Part of: Selfadjoint operator algebras Abstract harmonic analysis General theory of linear operators

Published online by Cambridge University Press:  14 February 2025

Charles Duquet  and
Christian Le Merdy
Charles Duquet
Affiliation:
Laboratoire de Mathématiques de Besançon, Université Marie et Louis Pasteur, Besançon, France
Christian Le Merdy*
Affiliation:
Laboratoire de Mathématiques de Besançon, Université Marie et Louis Pasteur, Besançon, France
*
Corresponding author: Christian Le Merdy; Email: clemerdy@univ-fcomte.fr
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Abstract

Let $\varphi$ be a normal semifinite faithful weight on a von Neumann algebra $A$, let $(\sigma ^\varphi _r)_{r\in {\mathbb R}}$ denote the modular automorphism group of $\varphi$, and let $T\colon A\to A$ be a linear map. We say that $T$ admits an absolute dilation if there exists another von Neumann algebra $M$ equipped with a normal semifinite faithful weight $\psi$, a $w^*$-continuous, unital and weight-preserving $*$-homomorphism $J\colon A\to M$ such that $\sigma ^\psi \circ J=J\circ \sigma ^\varphi$, as well as a weight-preserving $*$-automorphism $U\colon M\to M$ such that $T^k={\mathbb {E}}_JU^kJ$ for all integer $k\geq 0$, where $ {\mathbb {E}}_J\colon M\to A$ is the conditional expectation associated with $J$. Given any locally compact group $G$ and any real valued function $u\in C_b(G)$, we prove that if $u$ induces a unital completely positive Fourier multiplier $M_u\colon VN(G) \to VN(G)$, then $M_u$ admits an absolute dilation. Here, $VN(G)$ is equipped with its Plancherel weight $\varphi _G$. This result had been settled by the first named author in the case when $G$ is unimodular so the salient point in this paper is that $G$ may be nonunimodular, and hence, $\varphi _G$ may not be a trace. The absolute dilation of $M_u$ implies that for any $1\lt p\lt \infty$, the $L^p$-realization of $M_u$ can be dilated into an isometry acting on a noncommutative $L^p$-space. We further prove that if $u$ is valued in $[0,1]$, then the $L^p$-realization of $M_u$ is a Ritt operator with a bounded $H^\infty$-functional calculus.

Keywords

Fourier multipliersDilationsNon-commutative Lp-spaces

MSC classification

Primary: 47A20: Dilations, extensions, compressions 46L51: Noncommutative measure and integration 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 67 , Issue 3 , September 2025 , pp. 307 - 324
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Akcoglu, M. A., A pointwise ergodic theorem in Lp-spaces, Canadian J. Math. 27(5) (1975), 1075–1082.CrossRefGoogle Scholar
Akcoglu, M. A. and Sucheston, L., Dilations of positive contractions on Lp-spaces, Canad. Math. Bull. 20(3) (1977), 285–292.CrossRefGoogle Scholar
Anantharaman-Delaroche, C., On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Related Fields 135(4) (2006), 520–546.CrossRefGoogle Scholar
Arhancet, C., On Matsaev’s conjecture for contractions on noncommutative Lp-spaces, J. Operator Theory 69(2) (2013), 387–421.CrossRefGoogle Scholar
Arhancet, C., Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces, J. Funct. Anal. 276(7) (2019), 2279–2314.CrossRefGoogle Scholar
Arhancet, C., Dilations of Markovian semigroups of Fourier multipliers on locally compact groups, Proc. Amer. Math. Soc. 148(6) (2020), 2551–2563.CrossRefGoogle Scholar
Arhancet, C. and Merdy, C. Le, Dilation of Ritt operators on Lp-spaces, Israel J. Math. 201(1) (2014), 373–414.CrossRefGoogle Scholar
Arhancet, C., Fackler, S. and Merdy, C. Le, Isometric dilations and H∞-calculus for bounded analytic semigroups and Ritt operators, Trans. Amer. Math. Soc. 369(10) (2017), 6899–6933.CrossRefGoogle Scholar
Bergh, J. and Löfström, J., Interpolation spaces, an introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, (Springer-Verlag, Berlin-New York, 1976), x+207.Google Scholar
Blunck, S., Analyticity and discrete maximal regularity on Lp-spaces, J. Funct. Anal. 183(1) (2001), 211–230.CrossRefGoogle Scholar
Bozejko, M., Kümmerer, B. and Speicher, R., q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185(1) (1997), 129–154.Google Scholar
Bratelli, O. and Robinson, D., Operator algebras and quantum statistical mechanics II, Texts and Monographs in Physics (Springer-Verlag, New York-Berlin, 1981), xiii+505.Google Scholar
Caspers, M. and de la Salle, M., Schur and Fourier multipliers of an amenable group acting on non-commutative Lp-spaces, Trans. Amer. Math. Soc. 367(10) (2015), 6997–7013.CrossRefGoogle Scholar
De Cannière, J. and Haagerup, U., Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math 107(2) (1985), 455–500.CrossRefGoogle Scholar
Dungey, N., Subordinated discrete semigroups of operators, Trans. Amer. Math. Soc. 363(4) (2011), 1721–1741.CrossRefGoogle Scholar
Duquet, C., Dilation properties of measurable Schur multipliers and Fourier multipliers, Positivity 26(4) (2022), 41. Paper No. 69.CrossRefGoogle Scholar
Duquet, C. and Merdy, C. Le, A characterization of absolutely dilatable Schur multipliers, Adv. Math. 439(2024), 33. Paper No. 109492.CrossRefGoogle Scholar
Goldstein, S. and Lindsay, J. M., Markov semigroups KMS-symmetric for a weight, Math. Ann. 313(1) (1999), 39–67.CrossRefGoogle Scholar
Gomilko, A. and Tomilov, Y., On discrete subordination of power bounded and Ritt operators, Indiana Univ. Math. J. 67(2) (2018), 781–829.CrossRefGoogle Scholar
Haagerup, U., Lp-spaces associated with an abritrary von Neumann algebra, in Algèbresd’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), (Colloq. Internat. CNRS, 274, CNRS, Paris, 1979), 175–184.Google Scholar
Haagerup, U., On the dual weights for crossed products of von Neumann algebras II, Math. Scand 43 (1978), 119–140.CrossRefGoogle Scholar
Haagerup, U. and Musat, M., Factorization and dilation problems for completely positive maps on von Neumann algebras, Comm. Math. Phys. 303(2) (2011), 555–594.CrossRefGoogle Scholar
Haagerup, U., Junge, M. and Xu, Q., A reduction method for noncommutative Lp-spaces and applications, Trans. Amer. Math. Soc. 362(4) (2010), 2125–2165.CrossRefGoogle Scholar
Haak, B., Remarks on Ritt operators, their H∞-functional calculus and associated square function estimates, Preprint, arXiv: 2303.03022Google Scholar
Haak, B. and Haase, M., Square function estimates and functional calculi, Preprint, arXiv: 1311.0453Google Scholar
Hiai, F., Lectures on selected topics in von Neumann algebras, EMS Series of Lectures in Mathematics, (EMS Press, Berlin, 2021), viii+241CrossRefGoogle Scholar
Hong, G., Ray, S. and Wang, S., Maximal ergodic inequalities for some positive operators on noncommutative Lp-spaces, (1907), J. Lond. Math. Soc. (2) 108(1) (2023), 362–408.Google Scholar
Hytönen, T., van Neerven, J., Veraar, M. and Weis, L., Analysis in banach spaces, Probabilistic Methods and Operator Theory, vol. II (Springer, Cham, 2017), xxi+616. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 67Google Scholar
Junge, M. and Merdy, C. Le, Dilations and rigid factorisations on noncommutative Lp-spaces, J. Funct. Anal. 249(1) (2007), 220–252.CrossRefGoogle Scholar
Junge, M., Le Merdy, C.and Xu, Q., H∞ functional calculus and square functions on noncommutative Lp-spaces, Soc. Math. France, Astérisque 305 (2006), vi+138.Google Scholar
Kalton, N., Kunstmann, P. and Weis, L., Perturbation and interpolation theorems for the H∞-calculus with applications to differential operators, Math. Ann. 336(4) (2006), 747–801.CrossRefGoogle Scholar
Kosaki, H., Applications of the complex interpolation method to a von Neumann algebra: noncommutative Lp-spaces, J. Funct. Anal 56 (1984), 29–78.CrossRefGoogle Scholar
Kümmerer, B., Markov dilations on W*-algebras, J. Funct. Anal. 63(2) (1985), 139–177.CrossRefGoogle Scholar
Kunstmann, P. C. and Weis, L., Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems and H∞-functional calculus, functional analytic methods for evolution equations, in Lecture notes in math (Iannelli M., Nagel R. and Piazzera S., Editors) (Springer, Berlin, 2004). 1855Google Scholar
Merdy, C. Le, H∞- functional calculus and square function estimates for Ritt operators, Rev. Mat. Iberoam. 30(4) (2014), 1149–1190.CrossRefGoogle Scholar
Merdy, C. Le and Xu, Q., Maximal theorems and square functions for analytic operators on Lp-spaces, J. Lond. Math. Soc 2(2) (2012), 86-343–365.Google Scholar
Pisier, G. and Xu, Q., Non-Commutative -Spaces, Handbook of the Geometry of Banach Spaces, vol. 2, (North-Holland, Amsterdam, 2003), 1459–1517Google Scholar
Ricard, É., A Markov dilation for self-adjoint Schur multipliers, Proc. Amer. Math. Soc. 136(12) (2008), 4365–4372.CrossRefGoogle Scholar
Sratila, S. V., Modular theory in operator algebras, Editura Academiei Republicii Socialiste Romnia. (Abacus Press, Tunbridge Wells, Bucharest, 1981), 492Google Scholar
Sunder, V. S., An invitation to von Neumann algebras, Universitext. (Springer-Verlag, New York, 1987), xiv+171CrossRefGoogle Scholar
Takesaki, M., Theory of operator algebras. I, vol. 124 (Springer-Verlag, New York-Heidelberg, 1979), vii+415CrossRefGoogle Scholar
Takesaki, M., Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, Operator Algebras and Non-commutative Geometry, 6, vol. 125, (Springer-Verlag, Berlin, 2003), xxii+518.Google Scholar
Takesaki, M., Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, Operator Algebras and Non-commutative Geometry, 8, vol. 127 (Springer-Verlag, Berlin, 2003), xxii+548.Google Scholar
Terp, M., Lp- spaces associated with von Neumann algebras (1981). Notes, Math. Institute, Copenhagen UniversityGoogle Scholar
Terp, M., Interpolation spaces between a von Neumann algebra and its predual, J. Operator Theory 8(2) (1982), 327–360.Google Scholar
Vitse, P., A band limited and Besov class functional calculus for Tadmor-Ritt operators Arch. Math. (Basel) 85(4) 2005, 374–385.CrossRefGoogle Scholar
Xu, Q., Private communicationGoogle Scholar