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. i 1/2 Mm ,
Abstract. We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group Γ and relative Toeplitz-Schur multipliers on Schatten-vonNeumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum Λ ⊆ Γ , the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.
For any von Neumann algebra M, the noncommutative Mazur map Mp,q fromIn analogy with the commutative case, we gather estimates showing that Mp,q is min{ p q , 1}-Hölder on balls.2010 Mathematics Subject Classification: 46L51; 47A30.
International audienceWe construct several examples of Hilbertian operator spaces with few completely bounded maps. In particular, we give an example of a separable $1$-Hilbertian operator space $X_0$ such that, whenever $X'$ is an infinite dimensional quotient of $X_0$, $X$ is a subspace of $X'$, and $T : X \raw X'$ is a completely bounded map, then $T = \lambda I_{X} + S$, where $S$ is compact Hilbert-Schmidt and $||S||_2/16 \leq ||S||_{cb} \leq ||S||_2$. Moreover, every infinite dimensional quotient of a subspace of $X_0$ fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property
Abstract. We give a formula for Markov dilation in the sense of Anantharaman-Delaroche for real positive Schur multipliers on B(H).The classical theory of semigroups has many applications and connections with ergodic theory, martingales and probability (see [12]). The recent developments of noncommutative integration in von Neumann provide analogues of these notions ([1], [5], [6]). For instance, classical Markov semigroups on a probability space are generalized to semigroups of unital completely positive maps preserving a given faithful state. It is natural to try to adapt techniques from the commutative theory to the noncommutative one. Dealing with C *
Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful
semifinite normal weight $\phi$ and $\mathcal{N}$ be a von Neumann subalgebra
of $\mathcal{M}$ such that the restriction of $\phi$ to $\mathcal{N}$ is
semifinite and such that $\mathcal{N}$ is invariant by the modular group of
$\phi$. Let $\mathcal{E}$ be the weight preserving conditional expectation from
$\mathcal{M}$ onto $\mathcal{N}$. We prove the following inequality:
\[\|x\|_p^2\ge\bigl
\|\mathcal{E}(x)\bigr\|_p^2+(p-1)\bigl\|x-\mathcal{E}(x)\bigr\|_p^2, \qquad
x\in L_p(\mathcal{M}),1
0$ such that for any free group $\mathbb{F}_n$ and any
$q\ge4-\varepsilon_0$, \[\|P_t\|_{2\to q}\le1\quad\Leftrightarrow\quad
t\ge\log{\sqrt{q-1}},\] where $(P_t)$ is the Poisson semigroup defined by the
natural length function of $ \mathbb{F}_n$.Comment: Published at http://dx.doi.org/10.1214/14-AOP990 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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