On a characterization of commutativity for $$C^*$$ C ∗ -algebras via gyrogroup operations

2016 ◽  
Vol 72 (2) ◽  
pp. 248-251 ◽  
Author(s):  
Toshikazu Abe ◽  
Osamu Hatori
Keyword(s):  
C Algebras

scholarly journals A characterization of semiprojectivity for commutative C *-algebras

2012 ◽  
Vol 105 (5) ◽  
pp. 1021-1046 ◽  
Author(s):  
Adam P. W. Sørensen ◽  
Hannes Thiel
Keyword(s):  
C Algebras

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 Characterizations of tripotents in JB*-triples

2006 ◽  
Vol 99 (1) ◽  
pp. 147 ◽  
Author(s):  
Remo V. Hügli
Keyword(s):  
Unit Ball ◽  
Partial Order ◽  
Facial Structure ◽  
Partial Isometries ◽  
C Algebras ◽  
Metric Characterization

The set $\mathcal{U}(A)$ of tripotents in a $\mathrm{JB}^*$-triple $A$ is characterized in various ways. Some of the characterizations use only the norm-structure of $A$. The partial order on $\mathcal{U}(A)$ as well as $\sigma$-finiteness of tripotents are described intrinsically in terms of the facial structure of the unit ball $A_1$ in $A$, i.e. without reference to the (pre-)dual of $A$. This extends similar results obtained in [6] and simplifies the metric characterization of partial isometries in $C^*$-algebras found in [1](cf. [8].

Stability of Real C*-Algebras

2011 ◽  
Vol 54 (4) ◽  
pp. 593-606
Author(s):  
Jeffrey L. Boersema ◽  
Efren Ruiz
Keyword(s):  
Factorization Property ◽  
C Algebras ◽  
The One ◽  
Af Algebras

AbstractWe will give a characterization of stable real C*-algebras analogous to the one given for complex C*-algebras by Hjelmborg and Rørdam. Using this result, we will prove that any real C*-algebra satisfying the corona factorization property is stable if and only if its complexification is stable. Real C*-algebras satisfying the corona factorization property include AF-algebras and purely infinite C*-algebras. We will also provide an example of a simple unstable C*-algebra, the complexification of which is stable.

scholarly journals Equivariant KK-Theory and the Continuous Rokhlin Property

2020 ◽  
Author(s):  
Eusebio Gardella
Keyword(s):  
Fixed Point ◽  
Crossed Product ◽  
Compact Groups ◽  
Crossed Products ◽  
Circle Actions ◽  
Technical Result ◽  
C Algebras ◽  
Equivariant Version ◽  
Universal Coefficient Theorem

Abstract We introduce and study the continuous Rokhlin property for actions of compact groups on $C^*$-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong $KK$-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the Universal Coefficient Theorem (UCT) is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. As an application of the case of ${{\mathbb{Z}}}_3$-actions, we answer a question of Phillips–Viola about algebras not isomorphic to their opposites. Our analysis of the $KK$-theory of the crossed product allows us to prove a ${{\mathbb{T}}}$-equivariant version of Kirchberg–Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are $KK^{{{\mathbb{T}}}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant $K$-theory. We moreover characterize the $KK^{{{\mathbb{T}}}}$-theoretical invariants that arise in this way. Finally, we identify a $KK^{{{\mathbb{T}}}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.

scholarly journals CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OF CERTAIN C*-ALGEBRAS

2020 ◽  
pp. 1-12
Author(s):  
BHARAT TALWAR ◽  
RANJANA JAIN
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hausdorff Space ◽  
Separable Hilbert Space ◽  
Locally Compact ◽  
C Algebras ◽  
Finite Sum ◽  
Locally Compact Hausdorff Space ◽  
Quotient Property

Abstract For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

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