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222
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 expectation ..."
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Cited by 64 (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].
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 47 (6 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)
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.
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.
shifts, Cuntz algebras and multiresolution wavelet analysis of scale N, Integral Equations Operator Theory 28
, 1997
"... Abstract. In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras ON, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L 2 (T) by ..."
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Cited by 30 (15 self)
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Abstract. In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras ON, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L 2 (T) by (Siξ)(z) = mi (z)ξ ( z N). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over L 2 (T). This is used to compare the usual scale2 theory of wavelets with the scaleN theory. Also some other representations of ON of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant. Contents
Ergodic actions of universal quantum groups on operator algebras, preprint. 19 Wassermann, A., Ergodic actions of compact groups on operator algebras III: Classification for SU(2), Invent
 Math
, 1988
"... Abstract. 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 th ..."
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Cited by 25 (4 self)
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Abstract. 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. 1.
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.