Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Lifting graph C∗$C^*$‐algebra maps to Leavitt path algebra maps

Lifting graph C∗$C^*$‐algebra maps to Leavitt path algebra maps Let ξ:C∗(E)→C∗(F)$\xi :C^*(E)\rightarrow C^*(F)$ be a unital ∗$*$‐homomorphism between simple purely infinite Cuntz–Krieger algebras of finite graphs. We prove that there exists a unital ∗$*$‐homomorphism ϕ:L(E)→L(F)$\phi :L(E)\rightarrow L(F)$ between the corresponding Leavitt path‐algebras such that ξ$\xi$ is homotopic to the map ϕ̂:C∗(E)→C∗(F)$\hat{\phi }:C^*(E)\rightarrow C^*(F)$ induced by completion. We show moreover that ϕ̂$\hat{\phi }$ is a homotopy equivalence in the C∗$C^*$‐algebraic sense if and only if ϕ$\phi$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz–Krieger algebras is homotopic to the completion of a unital algebraic homotopy equivalence. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Lifting graph C∗$C^*$‐algebra maps to Leavitt path algebra maps

Loading next page...
 
/lp/wiley/lifting-graph-c-c-algebra-maps-to-leavitt-path-algebra-maps-amAfCgQkdu

References (29)

Publisher
Wiley
Copyright
© 2022 London Mathematical Society.
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms.12686
Publisher site
See Article on Publisher Site

Abstract

Let ξ:C∗(E)→C∗(F)$\xi :C^*(E)\rightarrow C^*(F)$ be a unital ∗$*$‐homomorphism between simple purely infinite Cuntz–Krieger algebras of finite graphs. We prove that there exists a unital ∗$*$‐homomorphism ϕ:L(E)→L(F)$\phi :L(E)\rightarrow L(F)$ between the corresponding Leavitt path‐algebras such that ξ$\xi$ is homotopic to the map ϕ̂:C∗(E)→C∗(F)$\hat{\phi }:C^*(E)\rightarrow C^*(F)$ induced by completion. We show moreover that ϕ̂$\hat{\phi }$ is a homotopy equivalence in the C∗$C^*$‐algebraic sense if and only if ϕ$\phi$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz–Krieger algebras is homotopic to the completion of a unital algebraic homotopy equivalence.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Dec 1, 2022

There are no references for this article.