Unconditionality, Fourier multipliers and Schur multipliers

Let $G$ be an infinite locally compact abelian group. If $X$ is Banach space, we show that if every bounded Fourier multiplier $T$ on $L^2(G)$ has the property that T\ot Id_X is bounded on $L^2(G,X)$ then the Banach space $X$ is isomorphic to a Hilbert space. Moreover, if $1<p<\infty$, $p\not=2$, we prove that there exists a bounded Fourier multiplier on $L^p(G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions to determine if an operator space is completely isomorphic to an operator Hilbert space.Comment: minor corrections; 17 pages ; to appear in Colloquium Mathematicu

Positive contractive projections on noncommutative Lp\mathrm{L}^p-spaces

In this paper, we prove the first theorems on contractive projections on general noncommutative $\mathrm{L}^p$-spaces associated with non-type I von Neumann algebras where $1 < p < \infty$. This is the first progress on this topic since the seminal work of Arazy and Friedman [Memoirs AMS 1992] where the problem of the description of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces is explicitly raised. We describe and we characterize precisely the positive contractive projections on a noncommutative $\mathrm{L}^p$-space associated with a $\sigma$-finite von Neumann algebra and we connect the problem to the theory of $\mathrm{JW}^*$-algebras. More precisely, we are able to show that the range of such a projection is isometric to some complex interpolation space $(N,N_*)_{\frac{1}{p}}$ where $N$ is a $\mathrm{JW}^*$-algebra in large cases. Our surprisingly simple approach relies on non tracial Haagerup's noncommutative $\mathrm{L}^p$-spaces in an essential way, even in the case of a projection acting on a finite-dimensional Schatten space and is unrelated to the methods of Arazy and Friedman.Comment: 24 pages, submitte

Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

We connect the (completely bounded) local Coulhon-Varopoulos dimension to the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) $\mathrm{L}^p$-spaces associated to finite measure spaces. As a consequence, we are able to prove that a two-sided Gaussian estimate on a heat kernel explicitely determines the spectral dimension. Our simple approach can be used with a large number of examples in various areas: Riemmannian manifolds, Lie groups, sublaplacians, doubling metric measure spaces, elliptic operators and quantum groups, in a unified manner.Comment: 25 page

Contractively decomposable projections on noncommutative Lp\mathrm{L}^p-spaces

We continue our investigation on contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces, started in [Arh1] and whose the description is explicitely asked in the seminal and influential work of Arazy and Friedman [Memoirs AMS 1992]. We show that the range of a contractively decomposable projection on an arbitrary noncommutative $\mathrm{L}^p$-space is completely isometrically isomorphic to some kind of $\mathrm{L}^p$-ternary ring of operators. In addition, we introduce the notion of $n$-pseudo-decomposable map where $n$ is an integer and we essentially reduce the study of the contractively $n$-pseudo-decomposable projections on noncommutative $\mathrm{L}^p$-spaces to the study of weak* contractive projections on $\mathrm{W}^*$-ternary rings of operators. Our approach is independent of the one of Arazy and Friedman.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1909.0039

Dilations of markovian semigroups of measurable Schur multipliers

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of Markov measurable Schur multipliers acting on $\mathrm{B}(\mathrm{L}^2(\Omega))$, where $\Omega$ is a measure space, can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh's $\mathrm{H}^\infty$ functional calculus of the generators of these semigroups on the associated noncommutative $\mathrm{L}^p$-spaces.Comment: 9 pages. arXiv admin note: substantial text overlap with arXiv:1811.0578