arXiv subjects
Publications and source records attributed to Dimitri Shlyakhtenko.
Suppose that μis an arbitrary Borel measure on the complex plane with compact support and take c > 0. If Z is a DT(μ,c)-operator as defined by Dykema and Haagerup, then the microstates free entropy dimension of Z is 2
We compute the group of Morita self-equivalences (the Picard group) of a Poisson structure on an orientable surface, under the assumption that the degeneracies of the Poisson tensor are linear. The answer involves mapping class groups of surfaces, i.e., groups of isotopy classes of diffeomorphisms. We also show that the Picard group of these structures coincides with the group of outer Poisson automorphisms.
We define the notion of L^2 homology and L^2 Betti numbers for a tracial von Neumann algebra, or, more generally, for any involutive algebra with a trace. The definition of these invariants is obtained from the definition of L^2 homology for groups, using the ideas from the theory of correspondences. For the group algebra of a discrete group, our Betti numbers coincide with the L^2 Betti numbers of the group. We find a link between the first L^2 Betti number and free entropy dimension, which points to the non-vanishing of L^2 homology for the von Neumann algebra of a free group.
We give an general estimate for the non-microstates free entropy dimension $δ^{*}(X_{1},..., X_{n})$. If $X_{1},..., X_{n}$ generate a diffuse von Neumann algebra, we prove that $δ^{*}(X_{1},..., X_{n})\geq 1$. In the case that $X_{1},..., X_{n}$ are $q$-semicircular variables as introduced by Bozejko and Speicher and $q^{2}n<1$, we show that $δ^{*}(X_{1},..., X_{n})>1$. We also show that for $|q|<\sqrt{2}-1$, the von Neumann algebras generated by a finite family of $q$-Gaussian random variables satisfy a condition of Ozawa and are therefore solid: the relative commutant of any diffuse subalgebra must be hyperfinite. In particular, when these algebras are factors, they are prime and do not have property $Γ$.
We show that Ozawa's recent results on solid von Neumann algebras imply that there are free Araki-Woods factors, which fail to have free absorption. We also show that a free Araki-Woods factors $Γ(μ, n)$ associated to a measure and a multiplicity function $n$ may non-trivially depend on the multiplicity function.
We prove several versions of Grothendieck's Theorem for completely bounded linear maps $T\colon E \to F^*$, when E and F are operator spaces. We prove that if E,F are $C^*$-algebras, of which at least one is exact, then every completely bounded $T\colon E \to F^*$ can be factorized through the direct sum of the row and column Hilbert operator spaces. Equivalently T can be decomposed as $T=T_r+T_c$ where $T_r$ (resp. $T_c$) factors completely boundedly through a row (resp. column) Hilbert operator space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Blecher on the factorization of completely bounded bilinear forms on $C^*$-algebras. Moreover, our result holds more generally for any pair E,F of "exact" operator spaces. This yields a characterization of the completely bounded maps from a $C^*$-algebra (or from an exact operator space) to the operator Hilbert space OH. As a corollary we prove that, up to a complete isomorphism, the row and column Hilbert operator spaces and their direct sums are the only operator spaces E such that both E and its dual $E^*$ are exact. We also characterize the Schur multipliers which are completely bounded from the space of compact operators to the trace class.
We introduce a new invariant \mathcal{S}(M) for type III factors M with no almost-periodic weights. We compute this invariant for certain free Araki-Woods factors. We show that Connes' invariant τcannot distinguish all isomorphism classes of free Araki-Woods factors. We show that there exists a continuum of mutually non-isomorphic free Araki-Woods factors, each without almost-periodic weights.
Let $M$ be a $B$-probability space. Assume that $B$ itself is a $D$-probability space; then $M$ can be viewed as $D$-probability space as well. Let $X$ be in $M$. We look at the question of relating the properties of $X$ as $B$-valued random variable to its properties as $D$-valued random variable. We characterize freeness of $X$ from $B$ with amalgamation over $D$: (a) in terms of a certain factorization condition linking the $B$-valued and $D$-valued cumulants of $X$, and (b) for $D$ finite-dimensional, in terms of linking the $B$-valued and the $D$-valued Fisher information of $X$. We give an application to random matrices. For the second characterization we derive a new operator-valued description of the conjugate variable and introduce an operator-valued version of the liberation gradient.
By utilizing an irreducible inclusion of type III$_{q^{2}} $ factors coming from a free-product type action of the quantum group $ SU_{q}(2) $, we show that the free group factor $ L(\mathbb {F}_{\infty}) $ possesses irreducible subfactors of arbitrary index $ >4 $. Combined with earlier results of Radulescu, this shows that $ L(\mathbb {F}_{\infty}) $ has irreducible subfactors with any index value in $ \{4\cos ^{2}(π/n):n\geq 3\}\cup [4,+\infty) $.
We extend Voiculescu's microstates-free definitions of free Fisher information and free entropy to the non-tracial framework. We explain the connection between these quantities and free entropy with respect to certain completely positive maps acting on the core of the non-tracial non-commutative probability space. We give a condition on free Fisher information of an infinite family of variables, which guarantees factoriality of the von Neumann algebra they generate.
Let M be a B-probability space. Assume that B itself is a D-probability space; then M can be viewed as a D-probability space as well. Let X be in M. We characterize freeness of X from B with amalgamation over D in terms of a certain factorization condition linking the B-valued and D-valued R-transforms of X. We give an application to random matrices.
We introduce the concept of ``R-cyclic family'' of matrices with entries in a non-commutative probability space; the definition consists in asking that only the ``cyclic'' non-crossing cumulants of the entries of the matrices are allowed to be non-zero. Let A_{1}, ..., A_{s} be an R-cyclic family of d \times d matrices over a non-commutative probability space. We prove a convolution-type formula for the explicit computation of the joint distribution of A_{1}, ..., A_{s} (considered in M_{d} (\A) with the natural state), in terms of the joint distribution (considered in the original space) of the entries of the s matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula. Moreover, let A_{1}, ..., A_{s} be a family of d \times d matrices over a non-commutative probability space, let \D \subset M_{d} (\A) denote the algebra of scalar diagonal matrices, and let {\cal C} be the subalgebra of M_{d} (\A) generated by \{A_{1}, ..., A_{s} \} \cup \D. We prove that the R-cyclicity of A_{1}, ..., A_{s} is equivalent to a property of {\cal C} -- namely that {\cal C} is free from M_{d} (\C), with amalgamation over \D.
We show that for each (0<λ<1), the free Araki-Woods factor of type III(_λ) cannot be written as a tensor product of two diffuse von Neumann algebras (i.e., is prime), and does not contain a Cartan subalgebra.
We define an analog of Voiculescu's free entropy for n-tuples of unitaries (u_{1},...,u_{n}) in a tracial von Neumann algebra M, normalizing a unital diffuse abelian subalgebra B in M. Using this quantity, we define the free dimension δ_{0}(u_{1},..,u_{n}\btw B). This number depends on (u_{1},... ,u_{n}) only up ``orbit equivalence'' over B. In particular, if R is an measirable equivalence relation on [0,1] generated by n automorphisms α_{1},...,α_{n}, let u_{1},..., u_{n} be the unitaries implementing α_{1},..,α_{n} in the Feldman-Moore crossed product algebra M=W^{*}([0,1],R), and let B be the canonical copy of L^\infty functions on [0,1] inside M. In this way, we obtain an invariant δ(R)=δ_{0}(u_{1},...,u_{n},\btw B) of the equivalence relation (R). If R is treeable, δ(R) coincides with the cost C(R) of R in the sense of Gaboriau. For a general equivalence relation R posessing a finite graphing, δ(R)\leq C(R). Using the notion of free dimension, we define an dynamical entropy invariant for an automorphism of a measurable equivalence relation (or more generally of an r-discrete measure groupoid), and give examples.
Let H be a full Hilbert bimodule over a C*-algebra A. We show that the Cuntz-Pimsner C*-algebra associated to H is exact if and only if A is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact C*-algebras. In the case that A is a finite dimensional C*-algebra, we also show that the Brown-Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz-Pimsner algebra associated to an A,A Hilbert bimodule is zero.
We define and study a relative free entropy quantity, analogous in its properties to Voiculescu's relative free entropy Chi^*(...:B). Our definition uses matricial microstates, unlike his definition, which involves non-commutative Hilbert transform. We prove a change of variable formula and certain maximization results for our quantity. We also exhibit a connection between the free entropy of a matrix with operator entries, relative to the algebra of scalar matrices, with the free entropy of the entries of the matrix.