Asymptotic lifts of positive linear maps

Lifting endomorphisms to automorphisms

Normal endomorphisms of von Neumann algebras need not be extendable to automorphisms of a larger von Neumann algebra, but they always have asymptotic lifts. We describe the structure of endomorphisms and their asymptotic lifts in some detail, and apply those results to complete the identification of asymptotic lifts of unital completely positive linear maps on von Neumann algebras in terms of their minimal dilations to endomorphisms.

The asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map L: M --> M acting on a von Neumann algebra - or more generally a dual operator system - we show that there is a unique reversible system α: N --> N (i.e., a complete order automorphism αof a dual operator system N) that captures all of the asymptotic behavior of L, called the {\em asymptotic lift} of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n x n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is …

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。

完全正映射的准不变提升 for 群作用

Quasi-invariant lifts of completely positive maps for groupoid actions

本文研究了群作用下完全正映射的准不变提升，并构造了G作用的Busby不变量。

Lifting theorems for completely positive maps

We prove lifting theorems for completely positive maps going out of exact $C^\ast$-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if $\mathsf X$ is a second countable topological space, $\mathfrak A$ and $\mathfrak B$ are separable, nuclear $C^\ast$-algebras over $\mathsf X$, and the action of $\mathsf X$ on $\mathfrak A$ is continuous, then $E(\mathsf X; \mathfrak A, \mathfrak B) \cong KK(\mathsf X; \mathfrak A, \mathfrak B)$ naturally. As an application, we show that a separable, nuclear, strongly purely infinite $C^\ast$-algebra $\mathfrak…

完全正映射的渐近提升

Asymptotic lifting for completely positive maps

本文证明了完全正压缩线性映射的渐近提升的存在性，并在等变情形下给出了群顺从性的新刻画。