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Berezin transforms on noncommutative varieties in polydomains 2013 Functional Analysis
**Abstract:** Let Q be a set of polynomials in noncommutative indeterminates Zi,j, i∈{1,…,k}, j∈{1,…,ni}. In this paper, we study noncommutative varieties

VQ(H):={X={Xi,j}∈D(H):g(X)=0 for all g∈Q},

where D(H) is a regular polydomain in B(H)n1+⋯+nk and B(H) is the algebra of bounded linear operators on a Hilbert space H. Under natural conditions on Q, we show that there is a universal model S={Si,j} such that g(S)=0, g∈Q, acting on a subspace of a tensor product of full Fock spaces. We characterize the variety VQ(H) and its pure part in terms of the universal model and a class of completely positive linear maps. We obtain a characterization of those elements in VQ(H) which admit characteristic functions and prove that the characteristic function is a complete unitary invariant for the class of completely non-coisometric elements. We study the universal model S, its joint invariant subspaces and the representations of the universal operator algebras it generates: the variety algebra A(VQ), the Hardy algebra F∞(VQ), and the C⁎-algebra C⁎(VQ). Using noncommutative Berezin transforms associated with each variety, we develop an operator model theory and dilation theory for large classes of varieties in noncommutative polydomains. This includes various commutative cases which are closely connected to the theory of holomorphic functions in several complex variables and algebraic geometry.

On the operator-valued analogues of the semicircle, arcsine and Bernoulli laws 2013 Functional AnalysisWe study the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions, respectively. In scalar-valued non-commutative probability these distributions are known to satisfy certain arithmetic relations with respect to Boolean and free convolutions. We show that, generally, the corresponding operator-valued distributions satisfy the same relations only when we consider them in the fully matricial sense introduced by Voiculescu. In addition, we provide a combinatorial description in terms of moments of the operator valued arcsine distribution and we show that its reciprocal Cauchy transform satisfies a version of the Abel equation similar to the one satisfied in the scalar-valued case.

Real second order freeness and Haar orthogonal matrices 2013 Functional Analysis**Abstract:** We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier.