arXiv subjects
Publications and source records attributed to Eric Ricard.
We investigate the notion of quasi-invariant states introduced in [2, 3] from an analytic viewpoint.We give the structures of quasi-invariant states with uniformly bounded cocycles. As a consequence, we can apply a Theorem of Kovacs and Szucs to get a conditional expectation on fixed points and another of Stormer to get an invariant semifinite trace under extra assumptions.
We prove a basic inequality involving anticommutators in noncommutative $L_p$-spaces. We use it to complete our study of the noncommutative Mazur maps from $L_p$ to $L_q$ showing that they are Lipschitz on balls when $0<q<p<\infty$.
We give a method to obtain, from Voiculescu's inequality, norm estimates for sums of free variables with amalgamation in general fully symmetric spaces. We use these estimates to interpolate the Burkholder inequalities for non commutative martingales. The method is also applicable to other similar settings. In that spirit, we improve known results on the non commutative Johnson-Schechtman inequalities and recover Khintchine inequalities associated to free groups.
We prove that the homogeneous functional calculus associated to $x\mapsto |x|^θ$ or $x\mapsto {\rm sgn}\, (x) |x|^θ$ for $0<θ<1$ is $θ$-Hölder on selfadjoint elements of noncommutative $L_p$-spaces for $0<p\leq\infty$ with values in $L_{p/θ}$. This extends an inequality of Birman, Koplienko and Solomjak also obtained by Ando.
We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1<p<\infty$. This implies that the decomposition of the free group $\F_\infty$ into reduced words starting with distinct free generators is completely unconditional in $L^p$. We study the case of Voiculescu's amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness-problem posed by Ozawa, a length independent estimate for Junge-Parcet-Xu's free Rosenthal inequality, a Littlewood-Paley-Stein type inequality for geodesic paths of free groups, and a length reduction formula for $L^p$-norms of free group von Neumann algebras.
We prove Khintchine type inequalities for words of a fixed length in a reduced free product of $C^*$-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length $d$ is completely bounded with norm depending linearly on $d$. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema's theorem on the stability of exactness under the reduced free product for $C^*$-algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite dimensional $C^*$-algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak-$*$ CCAP. In the case of group $C^*$-algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable.
We give a proof of the Khintchine inequalities in non-commutative $L_p$-spaces for all $0< p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for the analogues of such random variables in free probability. We also prove a factorization for operators from a Hilbert space to a non commutative $L_p$-space, which is new for $0<p<1$. We end by showing that Mazur maps are Hölder on semifinite von Neumann algebras. The main tool is a new form of Hölder inequality for non commutative Lp spaces with weights.
We prove that von Neumann algebras and separable nuclear $C^*$-algebras are stable for the Banach-Mazur cb-distance. A technical step is to show that unital almost completely isometric maps between $C^*$-algebras are almost multiplicative and almost selfadjoint. Also as an intermediate result, we compare the Banach-Mazur cb-distance and the Kadison-Kastler distance. Finally, we show that if two $C^*$-algebras are close enough for the cb-distance, then they have at most the same length.
We show that all the free Araki-Woods factors $\Gamma(H_\R, U_t)"$ have the complete metric approximation property. Using Ozawa-Popa's techniques, we then prove that every nonamenable subfactor $\mathcal{N} \subset \Gamma(H_\R, U_t)"$ which is the range of a normal conditional expectation has no Cartan subalgebra. We finally deduce that the type ${\rm III_1}$ factors constructed by Connes in the '70s can never be isomorphic to any free Araki-Woods factor, which answers a question of Shlyakhtenko and Vaes.
We prove that the von Neumann algebras generated by $n$ $q$-Gaussian elements, are factors for $n\ge 2$.
We consider the Schatten spaces S^p in the framework of operator space theory and for any $1\leq p\not=2<\infty$, we characterize the completely 1-complemented subspaces of S^p. They turn out to be the direct sums of spaces of the form S^p(H,K), where H,K are Hilbert spaces. This result is related to some previous work of Arazy-Friedman giving a description of all 1-complemented subspaces of S^p in terms of the Cartan factors of types 1-4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L^p-spaces. Also we show that for any $n\geq 2$, there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2n which is not completely isometric, and we investigate L^p-versions of such isomorphisms.
We consider the unitary group $\U$ of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever $\U$ acts by isometries on a metric space, every orbit is bounded. Equivalently, $\U$ is not the union of a countable chain of proper subgroups, and whenever $\E\subseteq \U$ generates $\U$, it does so by words of a fixed finite length.
We study the $t$-deformation of gaussian von Neumann algebras. They appear as example in the theories of Interacting Fock spaces and conditionally free products. When the number of generators is fixed, it is proved that if $t$ sufficiently close to 1, then these algebras do not depend on $t$. In the same way, the notion of conditionally free von Neumann algebras often coincides with freeness.
We prove a conjecture of Wojtaszczyk that for $1\leq p<\infty$, $p\neq 2$, $H_p(\mathbbT)$ does not admit any norm one projections with dimension of the range finite and bigger than 1. This implies in particular that for $1\leq p<\infty$, $p\ne 2$, $H_p$ does not admit a Schauder basis with constant one.