Extensions of C*-algebras and translation invariant asymptotic homomorphisms

渐近同伦提升性质

An asymptotic homotopy lifting property

本文提出了渐近同伦提升性质，并证明该性质对半射影$C^*$-代数的序列归纳极限等可分$C^*$-代数成立，且在商映射满足近似分解性质时对任意可分$C^*$-代数亦成立。

同伦提升、渐近同态与追踪

Homotopy lifting, asymptotic homomorphisms, and traces

本文证明了同伦提升定理，指出若两个*同态在模下同伦且一个可提升为渐近同态，则另一个也可提升。并探讨了C*-代数的重要性质及同伦提升在追踪理论中的应用。

Asymptotically split extensions and E-theory

We show that the E-theory of Connes and Higson can be formulated in terms of C*-extensions in a way quite similar to the way in which the KK-theory of Kasparov can. The essential difference is that the role played by split extensions should be taken by asymptotically split extensions. We call an extension of a C*-algebra $A$ by a stable C*-algebra $B$ asymptotically split if there exists an asymptotic homomorphism consisting of right inverses for the quotient map. An extension is called semi-invertible if it can be made asymptotically split by adding another extension to it. Our main result is…

The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite von Neumann Algebras

A triple of finite von Neumann algebras $B\subseteq N\subseteq M$ is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators $\{u_λ\}_{λ\in Λ}$ in $B$ such that $$\lim_λ|\mathbb{E}}_B(xu_λy)-{\mathbb{E}}_B({\mathbb{E}}_N(x)u_λ{\mathbb{E}}_N(y))\|_2=0$$ for all $x,y\in M$. We prove that a triple of finite von Neumann algebras $B\subseteq N\subseteq M$ has the relative weak asymptotic homomorphism property if and only if $N$ contains the set of all $x\in M$ such that $Bx\subseteq \sum_{i=1}^n x_iB$ for a finite number of elements $x_1,...,x_n$ i…

The Connes-Higson construction is an isomorphism

Let $A$ be a separable $C^*$-algebra and $B$ a stable $C^*$-algebra containing a strictly positive element. We show that the group $\Ext(SA,B)$ of unitary equivalence classes of extensions of $SA$ by $B$, modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes-Higson construction gives rise to an isomorphism between $\Ext(SA,B)$ and the $E$-theory group $E(A,B)$ of homotopy classes of asymptotic homomorphisms from $S^2A$ to $B$.

O carater de Chern-Connes para C$^*$-sistemas dinamicos calculado em algumas algebras de operadores pseudodiferenciais

Given a C$^*$-dynamical system $(A, G, α)$ one defines a homomorphism, called the Chern-Connes character, that take an element in $K_0(A) \oplus K_1(A)$, the K-theory groups of the C$^*$-algebra $A$, and maps it into $H_{\mathbb{R}}^*(G)$, the real deRham cohomology ring of $G$. We explictly compute this homomorphism for the examples $(\overline{Ψ_{cl}^0(S^1)}, S^1, α)$ and $(\overline{Ψ_{cl}^0(S^2)}, SO(3), α)$, where $\overline{Ψ_{cl}^0(M)}$ denotes the C$^*$-algebra generated by the classical pseudodifferential operators of zero order in the manifold $M$ and $α$ the action of conjugation by…

On the KK-theory of strongly self-absorbing C*-algebras

Let $\Dh$ and $A$ be unital and separable $C^{*}$-algebras; let $\Dh$ be strongly self-absorbing. It is known that any two unital $^*$-homomorphisms from $\Dh$ to $A \otimes \Dh$ are approximately unitarily equivalent. We show that, if $\Dh$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\Dh$ is asymptotically inner. Moreover, the space of automorphisms of $\Dh$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,\A…

Dynamic Asymptotic Dimension of Translation Actions on Compact Lie Groups

We develop a method to bound the dynamic asymptotic dimension of isometric group actions $Γ\curvearrowright X$ in terms of the asymptotic dimension of a space of graphs similar to a box space of $Γ$ (which also determines finite-dimensionality), and a geometric property of $X$ related to the doubling dimension. We apply this method to describe the DAD of translation actions by finitely generated subgroups of compact Lie groups, characterize finite dimensionality of such actions, and consequently bound the nuclear dimension of $C^*$-algebras arising from such actions by amenable groups.

Extensions of the reduced group C*-algebra of a free product of amenable groups

We prove that the unitary equivalence classes of extensions of C*_r(G) by any sigma-unital stable C"-algebra, taken modulo extensions which split via an asymptotic homomorphism, form a group which can be calculated from the universal coefficient theorem of KK-theory when G is a free product of a countable collection of amenable groups.

The C*-algebras qA\otimes K and S^2A\otimes K are asymptotically equivalent

Let $A$ be a separable $C^*$-algebra. We prove that its stabilized second suspension $S^2A\otimes \mathcal K$ and the $C^*$-algebra $qA\otimes \mathcal K$ constructed by Cuntz in the framework of his picture of KK-theory are asymptotically equivalent. This means that there exist asymptotic morphisms from each to the other whose compositions are homotopic to the identity maps. This result yields an easy description of the natural transformation from KK-theory to E-theory. One more corollary is the following. T. Loring ([3]) proved that any asymptotic morphism from $\qC$ to any $C^*$-algebra $B$…

On Graded K-theory, Elliptic Operators and the Functional Calculus

Let $A$ be a graded C*-algebra. We characterize Kasparov's K-theory group $\hat{K}_0(A)$ in terms of graded *-homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.