scholarly journals On deformations of C∗-algebras by actions of Kählerian Lie groups

2016 ◽  
Vol 27 (03) ◽  
pp. 1650023 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Victor Gayral ◽  
Sergey Neshveyev ◽  
Lars Tuset
Keyword(s):  
Lie Groups ◽  
Deformation Quantization ◽  
Oscillatory Integrals ◽  
The Other ◽  
C Algebras ◽  
Negatively Curved

We show that two approaches to equivariant strict deformation quantization of C[Formula: see text]-algebras by actions of negatively curved Kählerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual 2-cocycles, are equivalent.

scholarly journals Addendum: On deformations of C∗-algebras by actions of Kählerian Lie groups

2019 ◽  
Vol 30 (11) ◽  
pp. 1992002
Author(s):  
Pierre Bieliavsky ◽  
Victor Gayral ◽  
Sergey Neshveyev ◽  
Lars Tuset
Keyword(s):  
Lie Groups ◽  
Oscillatory Integrals ◽  
The Other ◽  
C Algebras ◽  
Negatively Curved

We show that two approaches to equivariant deformation of C[Formula: see text]-algebras by actions of negatively curved Kählerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual [Formula: see text]-cocycles, are still equivalent despite the nonunitarity of our [Formula: see text]-cocycles.

scholarly journals The Howe-Moore property for real and $p$-adic groups

2011 ◽  
Vol 109 (2) ◽  
pp. 201 ◽  
Author(s):  
Raf Cluckers ◽  
Yves Cornulier ◽  
Nicolas Louvet ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Lie Groups ◽  
Ergodic Measure ◽  
Oscillatory Integrals ◽  
The Other ◽  
Unitary Representations ◽  
Compact Groups ◽  
Locally Compact ◽  
Relative Version ◽  
Closed Normal Subgroup ◽  
Measure Preserving Actions

We consider in this paper a relative version of the Howe-Moore property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or $p$-adic Lie groups, the pairs with the relative Howe-Moore property with respect to a closed, normal subgroup. This involves, in one direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of oscillatory integrals.

Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra Detlef Müller

2014 ◽  
Keyword(s):  
Harmonic Analysis ◽  
Finite Type ◽  
Maximal Function ◽  
Oscillatory Integrals ◽  
The Other ◽  
Riemannian Surface ◽  
Fourier Restriction ◽  
Newton Polyhedra ◽  
The Given ◽  
Finite Type Hypersurfaces

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a smooth, finite type hypersurface in R³ with Riemannian surface measure dσ‎). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.

scholarly journals On completeness of holomorphic principal bundles

1975 ◽  
Vol 56 ◽  
pp. 121-138 ◽  
Author(s):  
Shigeru Takeuchi
Keyword(s):  
Algebraic Group ◽  
Lie Groups ◽  
Abelian Variety ◽  
Lie Group ◽  
Basic Structure ◽  
Factor Group ◽  
The Other ◽  
Algebraic Subgroup ◽  
Points Of View ◽  
Lie Subgroup

In this paper we shall investigate the structure of complex Lie groups from function theoretical points of view. A. Morimoto proved in [10] that every connected complex Lie group G has the smallest closed normal connected complex Lie subgroup Ge, such that the factor group G/Ge is Stein. On the other hand there hold the following two basic structure theorems (A1) and (A2) for a connected algebraic group G (cf. [12]). (A1): G has the smallest normal algebraic subgroup N such that the factor group G/N is an affine algebraic group. Moreover N is a connected central subgroup. (A2): G has the unique maximal connected affine algebraic subgroup L, where L is normal and the factor group G/L is an abelian variety.

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

scholarly journals Orbit equivalence and rigidity of ergodic actions of Lie groups

1981 ◽  
Vol 1 (2) ◽  
pp. 237-253 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Invariant Measure ◽  
Euclidean Space ◽  
Lie Groups ◽  
Homogeneous Spaces ◽  
The Other ◽  
Rigidity Theorem ◽  
Simple Groups ◽  
Orbit Equivalence ◽  
Finite Invariant Measure ◽  
Lattice Subgroups

AbstractThe rigidity theorem for ergodic actions of semi-simple groups and their lattice subgroups provides results concerning orbit equivalence of the actions of these groups with finite invariant measure. The main point of this paper is to extend the rigidity theorem on one hand to actions of general Lie groups with finite invariant measure, and on the other to actions of lattices on homogeneous spaces of the ambient connected group possibly without invariant measure. For example, this enables us to deduce non-orbit equivalence results for the actions of SL (n, ℤ) on projective space, Euclidean space, and general flag and Grassman varieties.

scholarly journals Deformation quantization via Fell bundles

2001 ◽  
Vol 89 (1) ◽  
pp. 135 ◽  
Author(s):  
Beatriz Abadie ◽  
Ruy Exel
Keyword(s):  
Deformation Quantization ◽  
Lens Spaces ◽  
Cross Sectional ◽  
C Algebra ◽  
C Algebras ◽  
Commutative Algebras ◽  
Heisenberg Manifolds ◽  
The Cross ◽  
Special Case

A method for deforming $C^*$-algebras is introduced, which applies to $C^*$-algebras that can be described as the cross-sectional $C^*$-algebra of a Fell bundle. Several well known examples of non-commutative algebras, usually obtained by deforming commutative ones by various methods, are shown to fit our unified perspective of deformation via Fell bundles. Examples are the non-commutative spheres of Matsumoto, the non-commutative lens spaces of Matsumoto and Tomiyama, and the quantum Heisenberg manifolds of Rieffel. In a special case, in which the deformation arises as a result of an action of $\boldsymbol R^{2d}$, assumed to be periodic in the first $d$ variables, we show that we get a strict deformation quantization.

scholarly journals C*-algebras generated by isometries and true representations

2019 ◽  
Vol 38 (5) ◽  
pp. 215-232
Author(s):  
Mamoon Ahmed
Keyword(s):  
The Other ◽  
Lattice Ordered Group ◽  
Covariant Representation ◽  
Ordered Group ◽  
Two Stage ◽  
C Algebras ◽  
Ordered Groups ◽  
Lattice Ordered Groups

Let (G; P) be a quasi-lattice ordered group. In this paper we present a modied proof of Laca and Raeburn's theorem about the covariant isometric representations of amenable quasi-lattice ordered groups [7, Theorem 3.7], by following a two stage strategy. First, we construct a universal covariant representation for a given quasi-lattice ordered group (G; P) and show that it is unique. The construction of this object is new; we have not followed either Nica's approach in [10] or Laca and Raeburn's approach in [7], although all three objects are essentially the same. Our approach is a very natural one and avoids some of the intricacies of the other approaches. Then we show if (G; P) is amenable, true representations of (G; P) generate C-algebras which are canonically isomorphic to the universal object.

scholarly journals DEFORMATION QUANTIZATION OF CLASSICAL FIELDS

2001 ◽  
Vol 16 (14) ◽  
pp. 2533-2558 ◽  
Author(s):  
H. GARCÍA-COMPEÁN ◽  
J. F. PLEBAŃSKI ◽  
M. PRZANOWSKI ◽  
F. J. TURRUBIATES
Keyword(s):  
Gauge Theory ◽  
Deformation Quantization ◽  
The Other ◽  
Star Products ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Quantization Formalism ◽  
Classical Fields ◽  
Free Fields ◽  
Nontrivial Topology

We study the deformation quantization of scalar and Abelian gauge classical free fields. Stratonovich–Weyl quantizer, star products and Wigner functionals are obtained in field and oscillator variables. The Abelian gauge theory is particularly intriguing since the Wigner functional is factorized into a physical part and the other one containing the constraints only. Some effects of nontrivial topology within the deformation quantization formalism are also considered.

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