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229
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 65 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
The C ∗ algebras of arbitrary graphs
 Rocky Mountain J. Math
"... Abstract. To an arbitrary directed graph we associate a rowfinite directed graph whose C ∗algebra contains the C ∗algebra of the original graph as a full corner. This allows us to generalize results for C ∗algebras of rowfinite graphs to C ∗algebras of arbitrary graphs: the uniqueness theorem, ..."
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Cited by 47 (20 self)
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Abstract. To an arbitrary directed graph we associate a rowfinite directed graph whose C ∗algebra contains the C ∗algebra of the original graph as a full corner. This allows us to generalize results for C ∗algebras of rowfinite graphs to C ∗algebras of arbitrary graphs: the uniqueness theorem, simplicity criteria, descriptions of the ideals and primitive ideal space, and conditions under which a graph algebra is AF and purely infinite. Our proofs require only standard CuntzKrieger techniques and do not rely on powerful constructs such as groupoids, ExelLaca algebras, or CuntzPimsner algebras. 1.
Noncommutative interpolation and Poisson transforms
 Israel J. Math
"... Abstract. General results of interpolation (eg. NevanlinnaPick) by elements in the noncommutative analytic Toeplitz algebra F ∞ (resp. noncommutative disc algebra An) with consequences to the interpolation by bounded operatorvalued analytic functions in the unit ball of C n are obtained. Noncommu ..."
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Cited by 46 (9 self)
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Abstract. General results of interpolation (eg. NevanlinnaPick) by elements in the noncommutative analytic Toeplitz algebra F ∞ (resp. noncommutative disc algebra An) with consequences to the interpolation by bounded operatorvalued analytic functions in the unit ball of C n are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra F ∞ /J on Hilbert spaces, where J is any w ∗closed, 2sided ideal of F ∞ , are obtained and used to construct a w ∗continuous, F ∞ /J–functional calculus associated to row contractions T = [T1,..., Tn] when f(T1,..., Tn) = 0 for any f ∈ J. Other properties of the dual algebra F ∞ /J are considered. In [Po5], the second author proved the following version of von Neumann’s inequality for row contractions: if T1,..., Tn ∈ B(H) (the algebra of all bounded linear operators on the Hilbert space H) and T = [T1,...,Tn] is a contraction, i.e., ∑n ∗ i=1 TiTi ≤ IH, then for every polynomial p(X1,..., Xn) on n noncommuting indeterminates, (1)
Free Quasifree States
, 1997
"... To a real Hilbert space and a oneparameter group of orthogonal transformations we associate a C∗algebra which admits a free quasifree state. This construction is a freeprobability analog of the construction of quasifree states on the CAR and CCR algebras. We show that under certain conditions, o ..."
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Cited by 33 (7 self)
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To a real Hilbert space and a oneparameter group of orthogonal transformations we associate a C∗algebra which admits a free quasifree state. This construction is a freeprobability analog of the construction of quasifree states on the CAR and CCR algebras. We show that under certain conditions, our C∗algebras are simple, and the free quasifree states are unique. The corresponding von Neumann algebras obtained via the GNS construction are free analogs of the ArakiWoods factors. Such von Neumann algebras can be decomposed into free products of other von Neumann algebras. For nontrivial oneparameter groups, these von Neumann algebras are type III factors. In the case the oneparameter group is nontrivial and almostperiodic, we show that Connes’ Sd invariant completely classifies these algebras.
From endomorphisms to automorphisms and back: dilations and full corners
 J. London Math. Soc
"... Abstract. When S is a discrete subsemigroup of a discrete group G such that G = S −1 S, it is possible to extend circlevalued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective e ..."
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Cited by 25 (5 self)
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Abstract. When S is a discrete subsemigroup of a discrete group G such that G = S −1 S, it is possible to extend circlevalued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective endomorphisms of a C*algebra to actions of G by automorphisms of a larger C*algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost– Connes Hecke C*algebra from number theory is constructed explicitly as an application: it is the crossed product C0(Af) ⋊ Q ∗ +, corresponding to the multiplicative action of the positive rationals on the additive group Af of finite adeles.
Discrete product systems of Hilbert bimodules
 Pacific J. Math
"... A Hilbert bimodule is a right Hilbert module X over a C ∗algebra A together with a left action of A as adjointable operators on X. We consider families X = {Xs: s ∈ P} of Hilbert bimodules, indexed by a semigroup P, which are endowed with a multiplication which implements isomorphisms Xs ⊗A Xt → Xs ..."
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Cited by 24 (0 self)
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A Hilbert bimodule is a right Hilbert module X over a C ∗algebra A together with a left action of A as adjointable operators on X. We consider families X = {Xs: s ∈ P} of Hilbert bimodules, indexed by a semigroup P, which are endowed with a multiplication which implements isomorphisms Xs ⊗A Xt → Xst; such a family is a called a product system. We define a generalized CuntzPimsner algebra OX, and we show that every twisted crossed product of A by P can be realized as OX for a suitable product system X. Assuming P is quasilattice ordered in the sense of Nica, we analyze a certain Toeplitz extension Tcov(X) ofOX by embedding it in a crossed product BP ⋊τ,XP which has been “twisted ” by X; our main Theorem is a characterization of the faithful representations of BP ⋊τ,XP. Introduction.
Graph inverse semigroups, groupoids and their
 C ∗ algebras, J. Operator Theory
"... Abstract. There is now a substantial literature on graph C ∗algebras. Under a locally finite condition on a countable, directed graph, Kumjian, Pask, Raeburn, Renault showed that the C ∗algebra of the graph can be realized as the C ∗algebra of the path groupoid, i.e. the groupoid determined by th ..."
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Cited by 23 (2 self)
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Abstract. There is now a substantial literature on graph C ∗algebras. Under a locally finite condition on a countable, directed graph, Kumjian, Pask, Raeburn, Renault showed that the C ∗algebra of the graph can be realized as the C ∗algebra of the path groupoid, i.e. the groupoid determined by the infinite paths in the graph. In the present paper, we remove the local finiteness requirement. The path groupoid in the general context is obtained through the universal groupoid of a certain inverse semigroup associated with the graph. This inverse semigroup is called the graph inverse semigroup, and graph representations turn out to be just representations of this inverse semigroup. A certain reduction of the universal groupoid gives the path groupoid of the graph, and its C ∗algebra is isomorphic to the C ∗algebra of the graph. The unit space of the path groupoid contains the infinite paths of the graph, but also contains some finite paths. We show that, as in the locally finite case, the path groupoid is always amenable, and we give a groupoid proof of a recent theorem of W. Szymanski, characterizing when a graph C ∗algebra is simple. 1.
Ergodic actions of universal quantum groups on operator algebras
, 1998
"... We construct ergodic actions of compact quantum groups on C∗algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups. In particular, we construct: (1). an ergodic action of the compact quantum Au(Q) on the type III ..."
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Cited by 22 (4 self)
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We construct ergodic actions of compact quantum groups on C∗algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups. In particular, we construct: (1). an ergodic action of the compact quantum Au(Q) on the type IIIλ Powers factor Rλ for an appropriate positive Q ∈ GL(2, R); (2). an ergodic action of the compact quantum group Au(n) on the hyperfinite II1 factor R; (3). an ergodic action of the compact quantum group Au(Q) on the Cuntz algebra On for each positive matrix Q ∈ GL(n, C); (4). ergodic actions of compact quantum groups on the their homogeneous spaces, and an example of a nonhomogeneous classical space that admits an ergodic action of a compact quantum group.