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Search: All articles in the CJM digital archive with keyword purely infinite

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1. CJM Online first

Pasnicu, Cornel; Phillips, N. Christopher
The weak ideal property and topological dimension zero
Following up on previous work, we prove a number of results for C*-algebras with the weak ideal property or topological dimension zero, and some results for C*-algebras with related properties. Some of the more important results include: $\bullet$ The weak ideal property implies topological dimension zero. $\bullet$ For a separable C*-algebra~$A$, topological dimension zero is equivalent to ${\operatorname{RR}} ({\mathcal{O}}_2 \otimes A) = 0$, to $D \otimes A$ having the ideal property for some (or any) Kirchberg algebra~$D$, and to $A$ being residually hereditarily in the class of all C*-algebras $B$ such that ${\mathcal{O}}_{\infty} \otimes B$ contains a nonzero projection. $\bullet$ Extending the known result for ${\mathbb{Z}}_2$, the classes of C*-algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian $2$-groups. $\bullet$ If $A$ and $B$ are separable, one of them is exact, $A$ has the ideal property, and $B$ has the weak ideal property, then $A \otimes_{\mathrm{min}} B$ has the weak ideal property. $\bullet$ If $X$ is a totally disconnected locally compact Hausdorff space and $A$ is a $C_0 (X)$-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then $A$ also has the corresponding property (for topological dimension zero, provided $A$ is separable). $\bullet$ Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C*-algebras including all separable locally AH~algebras. $\bullet$ The weak ideal property does not imply the ideal property for separable $Z$-stable C*-algebras. We give other related results, as well as counterexamples to several other statements one might hope for.

Keywords:ideal property, weak ideal property, topological dimension zero, $C_0 (X)$-algebra, purely infinite C*-algebra

2. CJM 2012 (vol 65 pp. 481)

Ara, Pere; Dykema, Kenneth J.; Rørdam, Mikael
Correction of Proofs in "Purely Infinite Simple $C^*$-algebras Arising from Free Product Constructions'' and a Subsequent Paper
The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple $C^*$-algebras arising from free product constructions}, Canad. J. Math. 50 (1998), 323--341 and of Theorem 3.1 of K. J. Dykema, Purely infinite simple $C^*$-algebras arising from free product constructions, II, Math. Scand. 90 (2002), 73--86 are corrected.

Keywords:C*-algebras, purely infinite

3. CJM 2011 (vol 64 pp. 705)

Thomsen, Klaus
Pure Infiniteness of the Crossed Product of an AH-Algebra by an Endomorphism
It is shown that simplicity of the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism implies that the algebra is also purely infinite, provided only that the endomorphism leaves no trace state invariant and takes the unit to a full projection.

Keywords:purely infinite $C^*$-algebras, crossed products

4. CJM 2007 (vol 59 pp. 343)

Lin, Huaxin
Weak Semiprojectivity in Purely Infinite Simple $C^*$-Algebras
Let $A$ be a separable amenable purely infinite simple \CA which satisfies the Universal Coefficient Theorem. We prove that $A$ is weakly semiprojective if and only if $K_i(A)$ is a countable direct sum of finitely generated groups ($i=0,1$). Therefore, if $A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal G}\subset A$ satisfying the following: for any contractive positive linear map $L: A\to B$ (for any \CA $B$) with $ \|L(ab)-L(a)L(b)\|<\dt$ for $a, b\in {\mathcal G}$ there exists a homomorphism $h\from A\to B$ such that $ \|h(a)-L(a)\|<\ep$ for $a\in {\mathcal F}$.

Keywords:weakly semiprojective, purely infinite simple $C^*$-algebras
Categories:46L05, 46L80

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