Filling families and strong pure infiniteness

TL;DR

This paper introduces filling families with matrix diagonalization to refine pure infiniteness concepts, demonstrating that certain tensor products and crossed products preserve strong pure infiniteness in $C^*$-algebras.

Contribution

It develops the concept of filling families with matrix diagonalization, improving results on local pure infiniteness and criteria for strong pure infiniteness in tensor and crossed products.

Findings

Minimal tensor product of strongly purely infinite and exact $C^*$-algebras is strongly purely infinite.

Provides a criterion for strong pure infiniteness of crossed products by endomorphisms.

Confirms that certain nuclear Cuntz-Pimsner algebras are strongly purely infinite and absorb $\\mathcal{O}_\infty$.

Abstract

We introduce filling families with matrix diagonalization as a refinement of the work by R{\o}rdam and the first named author. As an application we improve a result on local pure infiniteness and show that the minimal tensor product of a strongly purely infinite $C^*$-algebra and a exact $C^*$-algebra is again strongly purely infinite. Our results also yield a sufficient criterion for the strong pure infiniteness of crossed products $A\rtimes_\varphi \mathbb{N}$ by an endomorphism $\varphi$ of $A$ (cf. Theorem 7.6). Our work confirms that the special class of nuclear Cuntz-Pimsner algebras constructed by Harnisch and the first named author consist of strongly purely infinite $C^*$-algebras, and thus absorb $\mathcal{O}_\infty$ tensorially.