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 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 The irreducibility of C*-algebras acting on Hilbert C*-modules

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2425-2433
Author(s):  
Runliang Jiang
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
C Algebra ◽  
C Algebras ◽  
The Difference

Let B be a C*-algebra, E be a Hilbert B module and L(E) be the set of adjointable operators on E. Let A be a non-zero C*-subalgebra of L(E). In this paper, some new kinds of irreducibilities of A acting on E are introduced, which are all the generalizations of those associated to Hilbert spaces. The difference between these irreducibilities are illustrated by a number of counterexamples. It is concluded that for a full Hilbert B-module, these irreducibilities are all equivalent if and only if the underlying C*-algebra B is isomorphic to the C*-algebra of all compact operators on a Hilbert space.

Factorization in the Invertible Group of a C*-Algebra

1997 ◽  
Vol 49 (6) ◽  
pp. 1188-1205 ◽  
Author(s):  
Michael J. Leen
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
Upper Bounds ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Identity Component ◽  
C Algebras ◽  
Number Of Factors ◽  
Finite Products

AbstractIn this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), characterize the group of finite products of elements of S. The particular C*-algebras studied in this paper are either unital purely infinite simple or of the form (A ⊗ K)+, where A is any C*-algebra and K is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents (1+ nilpotent), positive invertibles and symmetries (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.

Stabilized $C^*$-algebras constructed from symbolic dynamical systems

2000 ◽  
Vol 20 (3) ◽  
pp. 821-841 ◽  
Author(s):  
KENGO MATSUMOTO
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Tensor Product ◽  
Separable Hilbert Space ◽  
Dynamical Property ◽  
Compact Operators ◽  
Topological Conjugacy ◽  
Fock Spaces ◽  
C Algebras ◽  
Creation Operators

We construct stabilized $C^*$-algebras from subshifts by using the dynamical property of the symbolic dynamical systems. We prove that the construction is dynamical and the $C^*$-algebras are isomorphic to the tensor product $C^*$-algebras between the algebra of all compact operators on a separable Hilbert space and the $C^*$-algebras constructed from creation operators on sub-Fock spaces associated with the subshifts. We also prove that the gauge actions on the stabilized $C^*$-algebras are invariant for topological conjugacy as two-sided subshifts under some conditions. Hence, if two subshifts are topologically conjugate as two-sided subshifts, the associated stabilized $C^*$-algebras are isomorphic so that their K-groups are isomorphic.

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 Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

scholarly journals Construction and pure infiniteness of $C^*$-algebras associated with lambda-graph systems

2005 ◽  
Vol 97 (1) ◽  
pp. 73 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Hilbert Space ◽  
Bratteli Diagram ◽  
C Algebra ◽  
C Algebras ◽  
Shift Transformation ◽  
Labeled Graphs ◽  
Graph System ◽  
Creation Operators

A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [16] the author has introduced a $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ associated with a $\lambda$-graph system $\mathfrak{L}$ by using groupoid method as a generalization of the Cuntz-Krieger algebras. In this paper, we concretely construct the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ by using both creation operators and projections on a sub Fock Hilbert space associated with $\mathfrak{L}$. We also introduce a new irreducible condition on $\mathfrak{L}$ under which the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ becomes simple and purely infinite.

scholarly journals Partial automorphisms of stable C*-algebras and Hilbert C*-bimodules

2005 ◽  
Vol 79 (3) ◽  
pp. 391-398
Author(s):  
Kazunori Kodaka
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Equivalence Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Countably Infinite

AbstractLet A be a C*-algebra and K the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. In this note, we shall show that there is an isomorphism of a semigroup of equivalence classes of certain partial automorphisms of A ⊗ K onto a semigroup of equivalence classes of certain countably generated A-A-Hilbert bimodules.

CONTINUOUS–TRACE C*-ALGEBRAS NOT ISOMORPHIC TO THEIR OPPOSITE ALGEBRAS

2001 ◽  
Vol 12 (03) ◽  
pp. 263-275 ◽  
Author(s):  
N. CHRISTOPHER PHILLIPS
Keyword(s):  
Homotopy Equivalent ◽  
C Algebra ◽  
C Algebras ◽  
Continuous Trace

We give examples of locally trivial continuous-trace C *-algebra not isomorphic to their opposite algebras. Our examples include a unital C *-algebra which is both stably isomorphic to and homotopy equivalent to its opposite algebra, a unital C *-algebra which is homotopy equivalent to but not stably isomorphic to its opposite algebra, and a unital C *-algebra which is not even stably homotopy equivalent to its opposite algebra.

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