Noncommutative Logarithmic Sobolev Inequalities

We show that the logarithmic Sobolev inequality holds for an arbitrary hypercontractive semigroup {e-tP}t >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e<^>{-tP}\}_{t\ge 0}$$\end{document} acting on a noncommutative probability space (M,tau)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathcal {M}},\tau )$$\end{document}: & Vert;x & Vert;Lp(logL)ps(M)<= cp,s & Vert;Ps(x)& Vert;Lp(M),1<p<infinity,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert x\Vert _{L_p(\log L)<^>{ps}({\mathcal {M}})}\le c_{p,s}\Vert P<^>s(x)\Vert _{L_p({\mathcal {M}})},\quad 1<p<\infty , \end{aligned}$$\end{document}for every mean zero x and 0<s<infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s<\infty $$\end{document}. By selecting s=1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=1/2$$\end{document}, one can recover the p-logarithmic Sobolev inequality whenever the Riesz transform is bounded. Our inequality applies to numerous concrete cases, including Poisson semigroups for free groups, the Ornstein-Uhlenbeck semigroup for mixed Q-gaussian von Neumann algebras, the free product for Ornstein-Uhlenbeck semigroups etc. This provides a unified approach for functional analysis form of logarithmic Sobolev inequalities in general noncommutative setting.