scholarly journals A Lifting Characterization of Rfd C*-Algebras

2014 ◽  
Vol 115 (1) ◽  
pp. 85 ◽  
Author(s):  
Don Hadwin
Keyword(s):  
Hilbert Spaces ◽  
Infinite Cardinal ◽  
Free Products ◽  
C Algebra ◽  
C Algebras ◽  
The Family

We prove a conjecture of Terry Loring that characterizes separable RFD C*-algebras in terms of a lifting property. In addition we introduce and study generalizations of RFD algebras. If $k$ is an infinite cardinal, we say a C*-algebra is residually less than $k$ dimensional, if the family of representations on Hilbert spaces of dimension less than $k$ separates the points of the algebra. We give characterizations of this property and prove that this class is closed under free products in the nonunital category. For free products in the unital category, the results depend on the cardinal $k$.

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].

Non-Commutative Deformations of C(T2) and K-Theory

1997 ◽  
Vol 08 (05) ◽  
pp. 555-571
Author(s):  
Cristina Cerri
Keyword(s):  
Positive Element ◽  
Crossed Product ◽  
C Algebra ◽  
Basic Properties ◽  
C Algebras ◽  
The Family ◽  
K Theory

For each α ≥ 0, let Bα be the universal C*-algebra generated by unitary elements uα, vα and a self-adjoint hα such that ||hα|| ≤ α and [Formula: see text]. In this work we prove that the family {Bα}α ∈ [0,∞[ extend the family of soft torus with the same basic properties, i.e., the field of C*-algebras {Bα}α ∈ [0,α0] is continuous and each Bα is a crossed product of a C*-algebra homotopically equivalent to C(S1) by Z. We then show that the K-groups of Bα are isomorphic to Z ⊕ Z. Applying results from the theory of rotation algebras we prove that every positive element (n,m) in K0(Bα) satisfies |m|α ≤ 2πn. It follows that these C*-algebras are not all homotopically equivalent to each other, although they have the same K-groups.

Free group C*-algebras associated with ℓp

2014 ◽  
Vol 25 (07) ◽  
pp. 1450065 ◽  
Author(s):  
Rui Okayasu
Keyword(s):  
Free Group ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Linear Functionals ◽  
C Algebra ◽  
Tracial State ◽  
C Algebras ◽  
Positive Linear Functionals ◽  
The Ideal

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

scholarly journals Compact actions on C*-algebras

1980 ◽  
Vol 21 (2) ◽  
pp. 143-149
Author(s):  
Charles A. Akemann ◽  
Steve Wright
Keyword(s):  
Banach Algebra ◽  
Compact Operator ◽  
Weakly Compact ◽  
C Algebra ◽  
C Algebras ◽  
Compact Perturbations

In Section 33 of [2], Bonsall and Duncan define an elementtof a Banach algebratoact compactlyonif the mapa→tatis a compact operator on. In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elementsbof a C*-algebrafor which the mapsa→ba,a→ab,a→ab+ba,a→babare compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the C*-automorphisms ofwhich are weakly compact perturbations of the identity.

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.

scholarly journals Conditions for primitivity of unital amalgamated full free products of finite-dimensional C*-algebras

2014 ◽  
Vol 25 (09) ◽  
pp. 1450086
Author(s):  
Francisco Torres-Ayala
Keyword(s):  
Free Products ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Trivial Center

We consider amalgamated unital full free products of the form A1*D A2, where A1, A2 and D are finite-dimensional C*-algebras and there are faithful traces on A1 and A2 whose restrictions to D agree. We provide several conditions on the matrices of partial multiplicities of the inclusions D ↪ A1 and D ↪ A2 that guarantee that the C*-algebra A1*D A2 is primitive. If the ranks of the matrices of partial multiplicities are one or all entries are 0 or ≥ 2, we prove that the algebra A1*D A2 is primitive if and only if it has a trivial center.

ORTHOGONALITY PRESERVERS REVISITED

2009 ◽  
Vol 02 (03) ◽  
pp. 387-405 ◽  
Author(s):  
María Burgos ◽  
Francisco J. Fernández-Polo ◽  
Jorge J. Garcés ◽  
Antonio M. Peralta
Keyword(s):  
Linear Operator ◽  
Natural Product ◽  
Bounded Linear Operator ◽  
Complete Characterization ◽  
Bounded Linear ◽  
C Algebra ◽  
Jordan Homomorphism ◽  
C Algebras ◽  
Orthogonality Preservers

We obtain a complete characterization of all orthogonality preserving operators from a JB *-algebra to a JB *-triple. If T : J → E is a bounded linear operator from a JB *-algebra (respectively, a C *-algebra) to a JB *-triple and h denotes the element T**(1), then T is orthogonality preserving, if and only if, T preserves zero-triple-products, if and only if, there exists a Jordan *-homomorphism [Formula: see text] such that S(x) and h operator commute and T(x) = h•r(h) S(x), for every x ∈ J, where r(h) is the range tripotent of h, [Formula: see text] is the Peirce-2 subspace associated to r(h) and •r(h) is the natural product making [Formula: see text] a JB *-algebra. This characterization culminates the description of all orthogonality preserving operators between C *-algebras and JB *-algebras and generalizes all the previously known results in this line of study.

scholarly journals AN INTERNAL CHARACTERIZATION OF COMPLETE POSITIVITY FOR ELEMENTARY OPERATORS

2002 ◽  
Vol 45 (2) ◽  
pp. 285-300
Author(s):  
Richard M. Timoney
Keyword(s):  
Linear Operators ◽  
Type I ◽  
Complete Positivity ◽  
Elementary Operator ◽  
Bounded Linear ◽  
C Algebra ◽  
C Algebras ◽  
Metric Operator ◽  
Elementary Operators

AbstractComplete positivity of ‘atomically extensible’ bounded linear operators between $C^*$-algebras is characterized in terms of positivity of a bilinear form on certain finite-rank operators. In the case of an elementary operator on a $C^*$-algebra, the approach leads us to characterize k-positivity of the operator in terms of positivity of a quadratic form on a subset of the dual space of the algebra and in terms of a certain inequality involving factorial states of finite type I.As an application we characterize those $C^*$-algebras where every k-positive elementary operator on the algebra is completely positive. They are either k-subhomogeneous or k-subhomogeneous by antiliminal. We also give a dual approach to the metric operator space introduced by Arveson.AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 47B47; 47B65

MONOTONE OPERATOR FUNCTIONS ON C*-ALGEBRAS

2005 ◽  
Vol 16 (02) ◽  
pp. 181-196 ◽  
Author(s):  
HIROYUKI OSAKA ◽  
SERGEI SILVESTROV ◽  
JUN TOMIYAMA
Keyword(s):  
Classical Case ◽  
Linear Operators ◽  
Monotone Functions ◽  
Bounded Linear ◽  
C Algebra ◽  
C Algebras ◽  
Operator Monotone ◽  
Operator Functions ◽  
Operator Monotone Functions

The article is devoted to investigation of classes of functions monotone as functions on general C*-algebras that are not necessarily the C*-algebra of all bounded linear operators on a Hilbert space as in classical case of matrix and operator monotone functions. We show that for general C*-algebras the classes of monotone functions coincide with the standard classes of matrix and operator monotone functions. For every class we give exact characterization of C*-algebras with this class of monotone functions, providing at the same time a monotonicity characterization of subhomogeneous C*-algebras. We use this result to generalize characterizations of commutativity of a C*-algebra based on monotonicity conditions for a single function to characterizations of subhomogeneity. As a C*-algebraic counterpart of standard matrix and operator monotone scaling, we investigate, by means of projective C*-algebras and relation lifting, the existence of C*-subalgebras of a given monotonicity class.

DEDEKIND COMPLETIONS OF BOUNDED ARCHIMEDEAN ℓ-ALGEBRAS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250139 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
PATRICK J. MORANDI ◽  
BRUCE OLBERDING
Keyword(s):  
Inverse Limit ◽  
Stone Duality ◽  
Weierstrass Theorem ◽  
Hausdorff Spaces ◽  
Dedekind Completion ◽  
C Algebra ◽  
C Algebras ◽  
Baer Rings ◽  
Projective Covers

All algebras considered in this paper are commutative with 1. Let baℓ be the category of bounded Archimedean ℓ-algebras. We investigate Dedekind completions and Dedekind complete algebras in baℓ. We give several characterizations for A ∈ baℓ to be Dedekind complete. Also, given A, B ∈ baℓ, we give several characterizations for B to be the Dedekind completion of A. We prove that unlike general Gelfand-Neumark-Stone duality, the duality for Dedekind complete algebras does not require any form of the Stone–Weierstrass Theorem. We show that taking the Dedekind completion is not functorial, but that it is functorial if we restrict our attention to those A ∈ baℓ that are Baer rings. As a consequence of our results, we give a new characterization of when A ∈ baℓ is a C*-algebra. We also show that A is a C*-algebra if and only if A is the inverse limit of an inverse family of clean C*-algebras. We conclude the paper by discussing how to derive Gleason's theorem about projective compact Hausdorff spaces and projective covers of compact Hausdorff spaces from our results.

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