scholarly journals Gabor duality theory for Morita equivalent C∗-algebras

2020 ◽  
Vol 31 (10) ◽  
pp. 2050073 ◽  
Author(s):  
Are Austad ◽  
Mads S. Jakobsen ◽  
Franz Luef
Keyword(s):  
Duality Theory ◽  
Morita Equivalence ◽  
Gabor Frames ◽  
Duality Principle ◽  
Gabor Analysis ◽  
Matrix Algebras ◽  
C Algebras ◽  
Morita Equivalent ◽  
Twisted Group ◽  
Density Theorems

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

scholarly journals MORITA EQUIVALENCE FOR C*-ALGEBRAS WITH THE WEAK BANACH–SAKS PROPERTY. II

2007 ◽  
Vol 50 (1) ◽  
pp. 185-195
Author(s):  
Masaharu Kusuda
Keyword(s):  
Morita Equivalence ◽  
C Algebras ◽  
Morita Equivalent ◽  
Finite Dimensional

AbstractLet $C^*$-algebras $A$ and $B$ be Morita equivalent and let $X$ be an $A$–$B$-imprimitivity bimodule. Suppose that $A$ or $B$ is unital. It is shown that $X$ has the weak Banach–Saks property if and only if it has the uniform weak Banach–Saks property. Thus, we conclude that $A$ or $B$ has the weak Banach–Saks property if and only if $X$ does so. Furthermore, when $C^*$-algebras $A$ and $B$ are unital, it is shown that $X$ has the Banach–Saks property if and only if it is finite dimensional.

FRAMES IN HILBERT C*-MODULES AND MORITA EQUIVALENT C*-ALGEBRAS

2016 ◽  
Vol 59 (1) ◽  
pp. 1-10 ◽  
Author(s):  
MASSOUD AMINI ◽  
MOHAMMAD B. ASADI ◽  
GEORGE A. ELLIOTT ◽  
FATEMEH KHOSRAVI
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Discrete Spectrum ◽  
Compact Operators ◽  
Morita Equivalence ◽  
Hilbert Modules ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Continuous Trace

AbstractWe show that the property of a C*-algebra that all its Hilbert modules have a frame, in the case of σ-unital C*-algebras, is preserved under Rieffel–Morita equivalence. In particular, we show that a σ-unital continuous-trace C*-algebra with trivial Dixmier–Douady class, all of whose Hilbert modules admit a frame, has discrete spectrum. We also show this for the tensor product of any commutative C*-algebra with the C*-algebra of compact operators on any Hilbert space.

scholarly journals Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

2021 ◽  
Vol 127 (2) ◽  
pp. 317-336
Author(s):  
Kazunori Kodaka
Keyword(s):  
Discrete Group ◽  
Morita Equivalence ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Countable Discrete Group ◽  
Relative Commutant ◽  
Strong Morita Equivalence

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

scholarly journals A classic Morita equivalence result for Fell bundle $C^*$-algebras

2011 ◽  
Vol 108 (2) ◽  
pp. 251
Author(s):  
Marius Ionescu ◽  
Dana P. Dilliams
Keyword(s):  
Closed Subgroup ◽  
Morita Equivalence ◽  
Equivariant Map ◽  
C Algebra ◽  
C Algebras ◽  
Stability Group ◽  
Morita Equivalent ◽  
Special Case ◽  
Equivalence Result

We show how to extend a classic Morita Equivalence Result of Green's to the $C^*$-algebras of Fell bundles over transitive groupoids. Specifically, we show that if $p:{\mathcal B}\to G$ is a saturated Fell bundle over a transitive groupoid $G$ with stability group $H=G(u)$ at $u\in G^{(0)}$, then $C^* (G,{\mathcal B})$ is Morita equivalent to $C^*(H,{\mathcal C})$, where ${\mathcal C}={\mathcal B}_{| H}$. As an application, we show that if $p:{\mathcal B}\to G$ is a Fell bundle over a group $G$ and if there is a continuous $G$-equivariant map $\sigma:$ Prim $A\to G/H$, where $A=B(e)$ is the $C^*$-algebra of $\mathcal B$ and $H$ is a closed subgroup, then $C^*(G,{\mathcal B})$ is Morita equivalent to $C^* (H,{\mathcal C}^{I})$ where ${\mathcal C}^{I}$ is a Fell bundle over $H$ whose fibres are $A/I$-$A/I$-imprimitivity bimodules and $I=\bigcap\{ P:\sigma(P)=eH\}$. Green's result is a special case of our application to bundles over groups.

scholarly journals Sampling and periodization of generators of Heisenberg modules

2019 ◽  
Vol 30 (10) ◽  
pp. 1950051
Author(s):  
Mads S. Jakobsen ◽  
Franz Luef
Keyword(s):  
Vector Bundles ◽  
General Setting ◽  
Gabor Frames ◽  
Locally Compact ◽  
Noncommutative Torus ◽  
Gabor Analysis ◽  
Matrix Algebras ◽  
Finite Dimensional ◽  
The Matrix ◽  
Lca Groups

This paper considers generators of Heisenberg modules in the case of twisted group [Formula: see text]-algebras of closed subgroups of locally compact abelian (LCA) groups and how the restriction and/or periodization of these generators yield generators for other Heisenberg modules. Since generators of Heisenberg modules are exactly the generators of (multi-window) Gabor frames, our methods are going to be from Gabor analysis. In the latter setting, the procedure of restriction and periodization of generators is well known. Our results extend this established part of Gabor analysis to the general setting of LCA groups. We give several concrete examples where we demonstrate some of the consequences of our results. Finally, we show that vector bundles over an irrational noncommutative torus may be approximated by vector bundles for finite-dimensional matrix algebras that converge to the irrational noncommutative torus with respect to the module norm of the generators, where the matrix algebras converge in the quantum Gromov–Hausdorff distance to the irrational noncommutative torus.

scholarly journals On Morita's fundamental theorem for $C^*$-algebras

2001 ◽  
Vol 88 (1) ◽  
pp. 137 ◽  
Author(s):  
David P. Blecher
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebra ◽  
Closed Subspace ◽  
Morita Equivalence ◽  
Operator Space ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Strong Morita Equivalence

We give a solution, via operator spaces, of an old problem in the Morita equivalence of $C^*$-algebras. Namely, we show that $C^*$-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule. An operator module over a $C^*$-algebra $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ of operator modules. Our main result remains true with respect to subcategories of $OMOD$ which contain $HMOD$ and the $C^*$-algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a $C^*$-algebra argument using compactness of the quasistate space of a $C^*$-algebra, and lowersemicontinuity in the enveloping von Neumann algebra.

scholarly journals *-IDEALS AND FORMAL MORITA EQUIVALENCE OF *-ALGEBRAS

2001 ◽  
Vol 12 (05) ◽  
pp. 555-577 ◽  
Author(s):  
HENRIQUE BURSZTYN ◽  
STEFAN WALDMANN
Keyword(s):  
Minimal Element ◽  
Minimal Ideal ◽  
General Setting ◽  
Quadratic Extension ◽  
Morita Equivalence ◽  
Equivalent Representation ◽  
C Algebras ◽  
Morita Equivalent ◽  
Morita Invariant ◽  
Rings With Involution

Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of *-algebras over a ring [Formula: see text] which is the quadratic extension by i of an ordered ring [Formula: see text]. The goal of the present paper is twofold. First, we clarify the relationship between formal Morita equivalence, Ara's notion of Morita *-equivalence of rings with involution, and strong Morita equivalence of C*-algebras. Second, in the general setting of *-algebras over [Formula: see text], we define "closed" *-ideals as the ones occurring as kernels of *-representations of these algebras on pre-Hilbert spaces. These ideals form a lattice which we show is invariant under formal Morita equivalence. This result, when applied to Pedersen ideals of C*-algebras, recovers the so-called Rieffel correspondence theorem. The triviality of the minimal element in the lattice of closed ideals, called the "minimal ideal", is also a formal Morita invariant and this fact can be used to describe a large class of examples of *-algebras over [Formula: see text] with equivalent representation theory but which are not formally Morita equivalent. We finally compute the closed *-ideals of some *-algebras arising in differential geometry.

scholarly journals Homogeneous C*-algebras whose spectra are tori

1985 ◽  
Vol 38 (1) ◽  
pp. 9-39 ◽  
Author(s):  
Shaun Disney ◽  
Iain Raeburn
Keyword(s):  
Isomorphism Class ◽  
Homotopy Theory ◽  
Transformation Group ◽  
Crossed Products ◽  
C Algebras ◽  
Isomorphism Classes ◽  
Twisted Group

AbstractBy a theorem of Fell and Tomiyama-Takesaki, an N-homogeneous C*-algebra with spectrum X has the form Γ(E) for some bundle E over X with fibre MN(C), and its isomorphism class is determined by that of E and its pull-backs f*E along homeomorphisms f of X. We describe the homogeneous C*-algebras with spectrum T2 or T3 by classifying the MN-bundles over Tk using elementary homotopy theory. We then use our results to determine the isomorphism classes of a variety of transformation group C*-algebras, twisted group C*-algebras and more general crossed products.

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