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.

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.

APPROXIMATELY INNER FLOWS ON SEPARABLE C*-ALGEBRAS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 649-673 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Irreducible Representation ◽  
Von Neumann Algebra ◽  
Automorphism Groups ◽  
Unitary Groups ◽  
Type I ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Inner Automorphism ◽  
Uniformly Continuous

We present two types of result for approximately inner one-parameter automorphism groups (referred to as AI flows hereafter) of separable C*-algebras. First, if there is an irreducible representation π of a separable C*-algebra A such that π(A) does not contain non-zero compact operators, then there is an AI flow α such that π is α-covariant and α is far from uniformly continuous in the sense that α induces a flow on π(A) which has full Connes spectrum. Second, if α is an AI flow on a separable C*-algebra A and π is an α-covariant irreducible representation, then we can choose a sequence (hn) of self-adjoint elements in A such that αt is the limit of inner flows Ad eithn and the sequence π(eithn) of one-parameter unitary groups (referred to as unitary flows hereafter) converges to a unitary flow which implements α in π. This latter result will be extended to cover the case of weakly inner type I representations. In passing we shall also show that if two representations of a separable simple C*-algebra on a separable Hilbert space generate the same von Neumann algebra of type I, then there is an approximately inner automorphism which sends one into the other up to equivalence.

A CONTINUUM OF C*-NORMS ON (H) ⊗ (H) AND RELATED TENSOR PRODUCTS

2015 ◽  
Vol 58 (2) ◽  
pp. 433-443 ◽  
Author(s):  
NARUTAKA OZAWA ◽  
GILLES PISIER
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
C Algebras ◽  
Injective Tensor Product ◽  
Algebraic Tensor Product

AbstractFor any pair M, N of von Neumann algebras such that the algebraic tensor product M ⊗ N admits more than one C*-norm, the cardinal of the set of C*-norms is at least 2ℵ0. Moreover, there is a family with cardinality 2ℵ0 of injective tensor product functors for C*-algebras in Kirchberg's sense. Let ${\mathbb B}$=∏nMn. We also show that, for any non-nuclear von Neumann algebra M⊂ ${\mathbb B}$(ℓ2), the set of C*-norms on ${\mathbb B}$ ⊗ M has cardinality equal to 22ℵ0.

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 Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

scholarly journals On the uniform Kadec-Klee property with respect to convergence in measure

1995 ◽  
Vol 59 (3) ◽  
pp. 343-352 ◽  
Author(s):  
F. A. Sukochev
Keyword(s):  
Function Space ◽  
Symmetric Function ◽  
Lorentz Space ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Finite Measure ◽  
Operator Space ◽  
Von Neumann ◽  
Convergence In Measure ◽  
Uniform Monotonicity

AbstractLet E(0, ∞) be a separable symmetric function space, let M be a semifinite von Neumann algebra with normal faithful semifinite trace μ, and let E(M, μ) be the symmetric operator space associated with E(0, ∞). If E(0, ∞) has the uniform Kadec-Klee property with respect to convergence in measure then E(M, μ) also has this property. In particular, if LΦ(0, ∞) (ϕ(0, ∞)) is a separable Orlicz (Lorentz) space then LΦ(M, μ) (Λϕ (M, μ)) has the uniform Kadec-Klee property with respect to convergence in measure on sets of finite measure if and only if the norm of E(0, ∞) satisfies G. Birkhoff's condition of uniform monotonicity.

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 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.

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