Remarks on the extension group for purely infinite corona algebras

ABSTRACT

We study BDF theory in the context of (not necessarily simple) purely infinite corona algebras. Let $\mathcal{B}$ be a nonunital separable simple C*-algebra with standard regularity properties for which $\mathcal{C}\left(\mathcal{B}\right)$ is purely infinite (though not necessarily simple). Let $\mathcal{A}$ be a separable nuclear C*-algebra. We prove a BDF Voiculescu decomposition theorem for maps from $\mathcal{A}$ to $\mathcal{C}\left(\mathcal{B}\right)$, and use this to prove that ${\mathbf{E}\mathbf{x}\mathbf{t}}_{\sigma }(\mathcal{A},\mathcal{B})$ is a group. Then, restricting to the case $\mathcal{A}=C\left(X\right)$, for some compact metric space X, and $\mathcal{B}$ has continuous scale, we provide characterizations of the neutral element for $\mathbf{E}\mathbf{x}\mathbf{t}\left(C\right(X),\mathcal{B})$.

KEYWORDS:

• C*-algebras
• operator theory
• corona algebras
• Brown–Douglas–Fillmore theory
• K theory

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 47L30
• 47C15
• 19K35

Disclosure statement

No potential conflict of interest was reported by the author(s).