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1. CMB Online first
A short note on the continuous Rokhlin property and the universal coefficient theorem in E-theory Let $G$ be a metrizable compact group, $A$ a separable $\mathrm{C}^*$-algebra
and $\alpha\colon G\to\operatorname{Aut}(A)$ a strongly continuous action.
Provided that $\alpha$ satisfies the continuous Rokhlin property,
we show that the property of satisfying the UCT in $E$-theory
passes from $A$ to the crossed product $\mathrm{C}^*$-algebra $A\rtimes_\alpha
G$ and the fixed point algebra $A^\alpha$. This extends a similar
result by Gardella for $KK$-theory in the case of unital
$\mathrm{C}^*$-algebras,
but with a shorter and less technical proof. For circle actions
on separable, unital $\mathrm{C}^*$-algebras with the continuous Rokhlin
property, we establish a connection between the $E$-theory equivalence
class of $A$ and that of its fixed point algebra $A^\alpha$.
Keywords:Rokhlin property, UCT, KK-theory, E-theory, circle actions Categories:46L55, 19K35 |
2. CMB 2010 (vol 54 pp. 82)
Lefschetz Numbers for $C^*$-Algebras
Using Poincar\'e duality, we obtain a formula of Lefschetz type
that computes the Lefschetz number of an endomorphism of a separable
nuclear $C^*$-algebra satisfying Poincar\'e duality and the Kunneth
theorem. (The Lefschetz number of an endomorphism is the graded trace
of the induced map on $\textrm{K}$-theory tensored with $\mathbb{C}$, as in the
classical case.) We then examine endomorphisms of Cuntz--Krieger
algebras $O_A$. An endomorphism has an invariant, which is a
permutation of an infinite set, and the contracting and expanding
behavior of this permutation describes the Lefschetz number of the
endomorphism. Using this description, we derive a closed polynomial
formula for the Lefschetz number depending on the matrix $A$ and the
presentation of the endomorphism.
Categories:19K35, 46L80 |
3. CMB 2007 (vol 50 pp. 227)
AF-Skeletons and Real Rank Zero Algebras with the Corona Factorization Property Let $A$ be a stable, separable, real rank zero $C^{*}$-algebra, and
suppose that $A$ has an AF-skeleton with only finitely many extreme
traces.
Then the corona algebra ${\mathcal M}(A)/A$ is
purely infinite in the sense of Kirchberg and R\o rdam, which implies that
$A$ has the corona factorization property.
Categories:46L80, 46L85, 19K35 |