Lifting graph C*-algebra maps to Leavitt path algebra maps.

GUILLERMO CORTIÑAS

BULLETIN OF THE LONDON MATHEMATICAL SOCIETY

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2022

0024-6093

Let ξ:C∗(E)→C∗(F) be a unital ∗-homomorphism between simple purelyinfinite Cuntz-Krieger algebras of finite graphs. We prove that there exists aunital ∗-homomorphism ϕ:L(E)→L(F) between the corresponding Leavittpath-algebras such that ξ is homotopic to the map ϕ^:C∗(E)→C∗(F) induced by completion. We show moreover that ϕ^ is a homotopyequivalence in the C∗-algebraic sense if and only if ϕ is a homotopyequivalence in the algebraic, polynomial sense. We deduce, in particular, thatany isomorphism between simple purely infinite Cuntz-Krieger algebras ishomotopic to the completion of a unital algebraic homotopy equivalence.