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A module frame concept for Hilbert C*modules, in “The functional and harmonic analysis of wavelets and frames
 Proceedings of AMS Special Session on the Functional and Harmonic Analysis
, 1999
"... Abstract. The goal of the present paper is a short introduction to a general module frame theory in C*algebras and Hilbert C*modules. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*modules over unital C*algebras that possess orthonormal Hilbert bases, and ..."
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Cited by 10 (4 self)
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Abstract. The goal of the present paper is a short introduction to a general module frame theory in C*algebras and Hilbert C*modules. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*modules over unital C*algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and other bounded module operators with suitable ranges. We obtain frame representation and decomposition theorems, as well as similarity and equivalence results. The relative position of two and more frames in terms of being complementary or disjoint is investigated in some detail. In the last section some recent results of P. G. Casazza are generalized to our setting. The Hilbert space situation appears as a special case. For the details of most of the proofs we refer to our basic publication [8]. Frames serve as a replacement for bases in Hilbert spaces that guarantee canonical reconstruction of every element of the Hilbert space by the reconstruction formula, however, giving up linear independence of the elements of the generating frame sequence. They appear naturally as wavelet generated and WeylHeisenberg / Gabor frames since often sequences of this type do not become orthonormal or Riesz bases, [11, 2, 12]. Similarly, the concept of module frames has become a
Symmetric approximation of frames and bases in Hilbert spaces, preprint
, 1998
"... Abstract. We consider existence and uniqueness of symmetric approximation of frames by normalized tight frames and of symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. A crucial role is played by the HilbertSchmidt property of a certain operator related to the initial fra ..."
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Cited by 7 (1 self)
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Abstract. We consider existence and uniqueness of symmetric approximation of frames by normalized tight frames and of symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. A crucial role is played by the HilbertSchmidt property of a certain operator related to the initial frame or basis. Given a Hilbert space H and a linearly independent set of vectors {f} n 1 in H the GramSchmidt process is traditionally used as a means of creating an orthonormal set of vectors {µi} n 1 from {fi} n 1 in such a way that span{fi} k 1 = span{µi} k 1 for all 1 ≤ k ≤ n. This process is inherently orderdependent in that a reordering of {fi} n 1 generally results in an entirely new orthonormal set {µi} n 1. For some problems in quantum chemistry, it is desirable to treat the vectors {f1, f2,..., fn} simultaneously in a method that is orderindependent instead of successively as in the GramSchmidt process. It is also desirable in such applications to orthonormalize the set {fi} n 1 in such a way that the sum ∑n j=1 ‖µj − fj ‖ 2 is minimal. The resulting set {µi} n 1 is called the symmetric or Löwdin orthogonalization of {fi} n 1.
Characterizing C*algebras of compact operators by generic categorical properties of Hilbert C*modules
, 2006
"... Abstract. B. Magajna and J. Schweizer showed in 1997 and 1999, respectively, that C*algebras of compact operators can be characterized by the property that every normclosed (and coinciding with its biorthogonal complement, resp.) submodule of every Hilbert C*module over them is automatically an o ..."
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Cited by 4 (3 self)
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Abstract. B. Magajna and J. Schweizer showed in 1997 and 1999, respectively, that C*algebras of compact operators can be characterized by the property that every normclosed (and coinciding with its biorthogonal complement, resp.) submodule of every Hilbert C*module over them is automatically an orthogonal summand. We find out further generic properties of the category of Hilbert C*modules over C*algebras which characterize precisely the C*algebras of compact operators. In 1997 B. Magajna obtained the equivalence of the property of the category of Hilbert C*modules over a certain C*algebra A that any Hilbert C*submodule is automatically an orthogonal summand with the property of the C*algebra A of coefficients to admit a faithful ∗representation in some C*algebra of compact operators on some Hilbert space, cf. Theorem 2.1. In 1999 J. Schweizer was able to sharpen the argument replacing the Hilbert C*module property of B. Magajna by the property of the category of Hilbert C*modules over a certain C*algebra A that any Hilbert C*submodule which coincides with its biorthogonal complement is automatically an orthogonal summand, cf. Theorem 2.1. Later on in 2003 M. Kusuda published further results which indicate that in the majority of situations the Hilbert C*module property can be weakened merely requiring the KA(M)Asubbimodules of the Hilbert C*modules M to be always orthogonal summands, see [11, 12] for the details. Studying the work of B. Magajna and J. Schweizer C*algebras A of the form A = c0 ∑ α ⊕K(Hα) become of special interest, where the symbol K(Hα) denotes the C*algebra of all compact operators on some Hilbert space Hi, and the c0sum is either a finite blockdiagonal sum or a blockdiagonal sum with a c0convergence condition on the C*algebra
INJECTIVE AND PROJECTIVE HILBERT C*MODULES, AND C*ALGEBRAS OF COMPACT OPERATORS
, 2008
"... Abstract. We consider projectivity and injectivity of Hilbert C*modules in the categories of Hilbert C*(bi)modules over a fixed C*algebra of coefficients (and another fixed C*algebra represented as bounded module operators) and bounded (bi)module morphisms, either necessarily adjointable or ar ..."
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Cited by 1 (1 self)
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Abstract. We consider projectivity and injectivity of Hilbert C*modules in the categories of Hilbert C*(bi)modules over a fixed C*algebra of coefficients (and another fixed C*algebra represented as bounded module operators) and bounded (bi)module morphisms, either necessarily adjointable or arbitrary ones. As a consequence of these investigations, we obtain a set of equivalent conditions characterizing C*subalgebras of C*algebras of compact operators on Hilbert spaces in terms of general properties of Hilbert C*modules over them. Our results complement results recently obtained by B. Magajna, J. Schweizer and M. Kusuda. In particular, all Hilbert C*(bi)modules over C*algebras of compact operators on Hilbert spaces are both injective and projective in the categories we consider. For more general C*algebras we obtain classes of injective and projective Hilbert C*(bi)modules. The goal of this paper is to determine the injective and projective Hilbert C*modules over a fixed C*algebra, when one allows the maps between C*modules to be bounded. Most prior work on injectivity has focused on the case of contractive maps. To better understand our motivations and the distinction between this work and the work of others, we first review the concept of injectivity and some history of the subject. To give a definition of the term, injective, that is useful for our purposes, we need a category, consisting of objects that are sets and morphisms between them that are functions, and for each object, N,
OPERATORVALUED FREE FISHER INFORMATION OF RANDOM MATRICES
, 2006
"... Abstract. We study the operatorvalued free Fisher information of random matrices in an operatorvalued noncommutative probability space. We obtain a formula for Φ ∗ M2(B) (A, A ∗ , M2(B), η), where A ∈ M2(B) is a 2 × 2 operator matrix on B, and η is linear operators on M2(B). Then we consider a spe ..."
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Abstract. We study the operatorvalued free Fisher information of random matrices in an operatorvalued noncommutative probability space. We obtain a formula for Φ ∗ M2(B) (A, A ∗ , M2(B), η), where A ∈ M2(B) is a 2 × 2 operator matrix on B, and η is linear operators on M2(B). Then we consider a special setting: A is an operatorvalued semicircular matrix with conditional expectation covariance, and find that Φ ∗ B (c, c ∗ : B, id) = 2Index(E), where E is a conditional expectation of B onto D and c is a circular variable with covariance E. 1. Introduction and
Modular frames for Hilbert C*modules and symmetric approximation of frames
, 2000
"... We give a comprehensive introduction to a general modular frame construction in Hilbert C*modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*modules over un ..."
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We give a comprehensive introduction to a general modular frame construction in Hilbert C*modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*modules over unital C*algebras that admit an orthonormal modular Riesz basis. Interrelations and applications to classical frame theory are indicated. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. In particular, the existence and uniqueness of the closest (normalized) tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closest tight frames often contains a multiple of its symmetric orthogonalization.
THE GAP BETWEEN UNBOUNDED REGULAR OPERATORS
, 901
"... Abstract. We study and compare the gap and the Riesz topologies of the space of all unbounded regular operators on Hilbert C*modules. We show that the space of all bounded adjointable operators on Hilbert C*modules is an open dense subset of the space of all unbounded regular operators with respec ..."
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Abstract. We study and compare the gap and the Riesz topologies of the space of all unbounded regular operators on Hilbert C*modules. We show that the space of all bounded adjointable operators on Hilbert C*modules is an open dense subset of the space of all unbounded regular operators with respect to the gap topology. The restriction of the gap topology on the space of all bounded adjointable operators is equivalent with the topology which is generated by the usual operator norm. The space of regular selfadjoint Fredholm operators on Hilbert C*modules over the C*algebra of compact operators is pathconnected with respect to the gap topology, however, the result may not be true for some Hilbert C*modules. 1. Introduction. Hilbert C*modules are essentially objects like Hilbert spaces, except that inner product, instead of being complexvalued, takes its values in a C*algebra. The theory of these modules, together with bounded and unbounded operators, is not only rich and attractive in its own right but forms an infrastructure for some of the most important research topics
OPERATORVALUED FRAMES ON C*MODULES
, 707
"... * The first and third authors are partially supported by Taft Foundations, the second author is partially supported by an NSF grant. Abstract. Frames on Hilbert C*modules have been defined for unital C*algebras by Frank and Larson [5] and operatorvalued frames on a Hilbert space have been studied ..."
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* The first and third authors are partially supported by Taft Foundations, the second author is partially supported by an NSF grant. Abstract. Frames on Hilbert C*modules have been defined for unital C*algebras by Frank and Larson [5] and operatorvalued frames on a Hilbert space have been studied in [8]. The goal of this paper is to introduce operatorvalued frames on a Hilbert C*module for a σunital C*algebra. Theorem 1.4 reformulates the definition given in [5] in terms of a series of rankone operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operatorvalued frame are limits in the strict topology of a series in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operatorvalued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes an onetoone correspondence between Murrayvon Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operatorvalued frames and provides a parametrization of all Parseval operatorvalued frames on a given Hilbert C*module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.
THE STANDARD MODEL – THE COMMUTATIVE CASE: SPINORS, DIRAC OPERATOR AND DE RHAM ALGEBRA
, 2000
"... Abstract. The present paper is a short survey on the mathematical basics of Classical Field Theory including the SerreSwan ’ theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin Cstructures, Dirac operators, exterior algebra bundles and Connes ’ differential ..."
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Abstract. The present paper is a short survey on the mathematical basics of Classical Field Theory including the SerreSwan ’ theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin Cstructures, Dirac operators, exterior algebra bundles and Connes ’ differential algebras in the commutative case, among other elements. We avoid the introduction of principal bundles and put the emphasis on a modulebased approach using SerreSwan’s theorem, Hermitian structures and module frames. A new proof (due to Harald Upmeier) of the differential algebra isomorphism between the set of smooth sections of the exterior algebra bundle and Connes ’ differential algebra is presented. The content of the present paper reflects a talk given at the Workshop ’The Standard Model of Elementary Particle Physics from a mathematicalgeometrical viewpoint ’ held