Duality for Actions and Coactions of Measured Groupoids of Von Neumann Algebras
Duality for Actions and Coactions of Measured Groupoids of Von Neumann Algebras

Author: Takehiko Yamanouchi

Publisher: American Mathematical Soc.

Published: 1993-01-01

Total Pages: 124

ISBN-13: 9780821862070

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Through classification of compact abelian group actions on semifinite injective factors, Jones and Takesaki introduced the notion of an action of a measured groupoid on a von Neumann algebra, which has proven to be an important tool for this kind of analysis. Elaborating on this notion, this work introduces a new concept of a measured groupoid action that may fit more perfectly into the groupoid setting. Yamanouchi also shows the existence of a canonical coproduct on every groupoid von Neumann algebra, which leads to a concept of a coaction of a measured groupoid. Yamanouchi then proves duality between these objects, extending Nakagami-Takesaki duality for (co)actions of locally compact groups on von Neumann algebras.

Duality for Actions and Coactions of Measured Groupoids on von Neumann Algebras
Duality for Actions and Coactions of Measured Groupoids on von Neumann Algebras

Author: Takehiko Yamanouchi

Publisher: American Mathematical Soc.

Published: 1993

Total Pages: 122

ISBN-13: 0821825453

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Through classification of compact abelian group actions on semifinite injective factors, Jones and Takesaki introduced a notion of an action of a measured groupoid on a von Neumann algebra, which was proven to be an important tool for such an analysis. In this paper, elaborating their definition, the author introduces a new concept of a measured groupoid action that may fit more perfectly in the groupoid setting. The author also considers a notion of a coaction of a measured groupoid by showing the existence of a canonical "coproduct" on every groupoid von Neumann algebra.

Selfadjoint and Nonselfadjoint Operator Algebras and Operator Theory
Selfadjoint and Nonselfadjoint Operator Algebras and Operator Theory

Author: Robert S. Doran

Publisher: American Mathematical Soc.

Published: 1991

Total Pages: 242

ISBN-13: 0821851276

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This book contains papers presented at the NSF/CBMS Regional Conference on Coordinates in Operator Algebras, held at Texas Christian University in Fort Worth in May 1990. During the conference, in addition to a series of ten lectures by Paul S Muhly (which will be published in a CBMS Regional Conference Series volume), there were twenty-eight lectures delivered by conference participants on a broad range of topics of current interest in operator algebras and operator theory. This volume contains slightly expanded versions of most of those lectures. Participants were encouraged to bring open problems to the conference, and, as a result, there are over one hundred problems and questions scattered throughout this volume. Readers will appreciate this book for the overview it provides of current topics and methods of operator algebras and operator theory.

Crossed Products of von Neumann Algebras by Equivalence Relations and Their Subalgebras
Crossed Products of von Neumann Algebras by Equivalence Relations and Their Subalgebras

Author: Igor Fulman

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 122

ISBN-13: 0821805576

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In this book, the author introduces and studies the construction of the crossed product of a von Neumann algebra. This construction is the generalization of the construction of the crossed product of an abelian von Neumann algebra by an equivalence relation introduced by J. Feldman and C. C. Moore. Many properties of this construction are proved in the general case. In addition, the generalizations of the Spectral Theorem on Bimodules and of the theorem on dilations are proved.

Manifolds with Group Actions and Elliptic Operators
Manifolds with Group Actions and Elliptic Operators

Author: Vladimir I︠A︡kovlevich Lin

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 90

ISBN-13: 0821826042

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This work studies equivariant linear second order elliptic operators [italic capital]P on a connected noncompact manifold [italic capital]X with a given action of a group [italic capital]G. The action is assumed to be cocompact, meaning that [italic capitals]GV = [italic capital]X for some compact subset of [italic capital]V of [italic capital]X. The aim is to study the structure of the convex cone of all positive solutions of [italic capital]P[italic]u = 0.

Deformation Quantization for Actions of $R^d$
Deformation Quantization for Actions of $R^d$

Author: Marc Aristide Rieffel

Publisher: American Mathematical Soc.

Published: 1993

Total Pages: 110

ISBN-13: 0821825755

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This work describes a general construction of a deformation quantization for any Poisson bracket on a manifold which comes from an action of R ]d on that manifold. These deformation quantizations are strict, in the sense that the deformed product of any two functions is again a function and that there are corresponding involutions and operator norms. Many of the techniques involved are adapted from the theory of pseudo-differential operators. The construction is shown to have many favorable properties. A number of specific examples are described, ranging from basic ones such as quantum disks, quantum tori, and quantum spheres, to aspects of quantum groups.

Duality and Definability in First Order Logic
Duality and Definability in First Order Logic

Author: Michael Makkai

Publisher: American Mathematical Soc.

Published: 1993

Total Pages: 122

ISBN-13: 0821825658

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We develop a duality theory for small Boolean pretoposes in which the dual of the [italic capital]T is the groupoid of models of a Boolean pretopos [italic capital]T equipped with additional structure derived from ultraproducts. The duality theorem states that any small Boolean pretopos is canonically equivalent to its double dual. We use a strong version of the duality theorem to prove the so-called descent theorem for Boolean pretoposes which says that category of descent data derived from a conservative pretopos morphism between Boolean pretoposes is canonically equivalent to the domain-pretopos. The descent theorem contains the Beth definability theorem for classical first order logic. Moreover, it gives, via the standard translation from the language of categories to symbolic logic, a new definability theorem for classical first order logic concerning set-valued functors on models, expressible in purely syntactical (arithmetical) terms.

Unraveling the Integral Knot Concordance Group
Unraveling the Integral Knot Concordance Group

Author: Neal W. Stoltzfus

Publisher: American Mathematical Soc.

Published: 1977

Total Pages: 103

ISBN-13: 082182192X

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The group of concordance classes of high dimensional homotopy spheres knotted in codimension two in the standard sphere has an intricate algebraic structure which this paper unravels. The first level of invariants is given by the classical Alexander polynomial. By means of a transfer construction, the integral Seifert matrices of knots whose Alexander polynomial is a power of a fixed irreducible polynomial are related to forms with the appropriate Hermitian symmetry on torsion free modules over an order in the algebraic number field determined by the Alexander polynomial. This group is then explicitly computed in terms of standard arithmetic invariants. In the symmetric case, this computation shows there are no elements of order four with an irreducible Alexander polynomial. Furthermore, the order is not necessarily Dedekind and non-projective modules can occur. The second level of invariants is given by constructing an exact sequence relating the global concordance group to the individual pieces described above. The integral concordance group is then computed by a localization exact sequence relating it to the rational group computed by J. Levine and a group of torsion linking forms.

The Kinematic Formula in Riemannian Homogeneous Spaces
The Kinematic Formula in Riemannian Homogeneous Spaces

Author: Ralph Howard

Publisher: American Mathematical Soc.

Published: 1993

Total Pages: 82

ISBN-13: 0821825690

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This memoir investigates a method that generalizes the Chern-Federer kinematic formula to arbitrary homogeneous spaces with an invariant Riemannian metric, and leads to new formulas even in the case of submanifolds of Euclidean space.