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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 98 (3 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].
Wavelet representations and Fock space on positive matrices
 J. Funct. Anal
, 2003
"... Abstract. We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntzalgebra representations in that special case. Each of these representations is s ..."
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Cited by 15 (11 self)
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Abstract. We show that every biorthogonal wavelet determines a representation by operators on Hilbert space satisfying simple identities, which captures the established relationship between orthogonal wavelets and Cuntzalgebra representations in that special case. Each of these representations is shown to have tractable finitedimensional coinvariant doublycyclic subspaces. Further, motivated by these representations, we introduce a general Fockspace Hilbert space construction which yields creation operators containing the Cuntz–Toeplitz isometries as a special case. In this paper, we wish to establish a connection between biorthogonal wavelets on the one hand [16], and representation theory for operators on Hilbert space on the other [9, 18]. This is accomplished by showing that each of these wavelets yields a collection of operators acting on Hilbert space which satisfy simple identities, and which contain the Cuntz relations [15] as a special case. In fact, this new relationship collapses to the now wellknown connection between orthogonal wavelets and representations of the Cuntz C ∗algebra in that special case [10]. Our second goal is to develop a framework for studying this new class of representations. Toward this end, we introduce a general Fock space Hilbert space construction which reduces to unrestricted Fock space in the familiar cases. Indeed, the natural creation operators we get can be thought of as an analogue of the Cuntz–Toeplitz creation operators to this more general setting. We regard this construction and the creation operators determined by it as interesting objects of study in their own right. Finally, our hope is that this paper will lead to further study of the relationships and objects introduced here.
Harmonic Analysis Of Fractal Processes Via C*Algebras
 C algebras, Math. Nachr
, 1995
"... . We construct a harmonic analysis of iteration systems which include those which arise from wavelet algorithms based on multiresolutions. While traditional discretizations lead to asymptotic formulas, we argue here for a direct Fourier duality; but it is based on a noncommutative harmonic analysis ..."
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Cited by 6 (5 self)
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. We construct a harmonic analysis of iteration systems which include those which arise from wavelet algorithms based on multiresolutions. While traditional discretizations lead to asymptotic formulas, we argue here for a direct Fourier duality; but it is based on a noncommutative harmonic analysis, specifically on representations of the Cuntz C algebras. With this approach the scaling from the wavelet takes the form of an endomorphism of B (H), H a Hilbert space derived from the lattice of translations. We use this to describe, and to calculate, new invariants for the wavelets. For those iteration systems which arise from wavelets and from Julia sets, we show that the associated endomorphisms are in fact Powers shifts. 1. The Coding Space While automorphisms of measure spaces correspond to dynamical systems with (time) reversal, the irreversible (or dissipative) dynamical systems arise from endomorphisms of the measure space in question. The measure spaces considered here will ...
Multiresolution wavelets analysis of integer scale Bessel functions
 J. Math. Phys
"... theory, special functions, recurrence relations, Hilbert space. AIP classification: mathematical methods in physics 02.30.f, 02.30.Mv, 02.30.Tb, 02.30.Uu, 02.50.Cw We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived ..."
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Cited by 5 (5 self)
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theory, special functions, recurrence relations, Hilbert space. AIP classification: mathematical methods in physics 02.30.f, 02.30.Mv, 02.30.Tb, 02.30.Uu, 02.50.Cw We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbertspace considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the C ∗algebra Oν+1 arising from this multiresolution analysis. A connection with Markov chains and representations of Oν+1 is found. Projection valued measures arising from the multiresolution analysis give rise to a Markov trace for quantum groups SOq. 1 1