Linear maps preserving Kasparov cycles and the characterization of induced automorphisms

Abstract

Let $E$ be a Hilbert C*-module and $\mathcal{L}(E)$ the C*-algebra of all bounded adjointable operators on $E$. An operator $T \in \mathcal{L}(E)$ is a Kasparov cycle if $T^*T - 1$ and $TT^*-1$ are compact operators on $E$. If $\mathcal{L}(E)\rightarrow \mathcal{L}(E)$ is a prime C*-algebra, and $φ:\mathcal{L}(E)\rightarrow \mathcal{L}(E)$ is a linear map which is unital and surjective up to compact operators, and preserves Kasparov cycles in both directions, then the induced map $ψ:\mathcal{L}(E)/\mathcal{K}(E) \rightarrow \mathcal{L}(E)/\mathcal{K}(E)$ is either a $*$-automorphism or a $*$-anti-automorphism. Our work extends the main result of [J. Math. Anal. Appl. 354 (2009), 625-629] to the set of Kasparov cycles, showing that the assumption ``$\mathcal{L}(E)/\mathcal{K}(E)$ has real rank zero'' is redundant in our results.