Tensorial Permanence of $K$-Stability for Diagonal AH-Algebras

TL;DR

This paper characterizes when tensor products of diagonal AH-algebras with arbitrary C*-algebras are K-stable, linking it to the unbounded growth of matrix block sizes in the inductive system.

Contribution

It provides a precise criterion for K-stability of tensor products of diagonal AH-algebras, connecting it to matrix block size growth, and applies it to various classes of algebras.

Findings

K-stability of tensor products is characterized by unbounded matrix block sizes.

Non-𝒵-stable Villadsen algebras of the first kind are K-stable when tensored with any C*-algebra.

Simple, unital, infinite-dimensional diagonal AH-algebras are always K-stable when tensoring with arbitrary C*-algebras.

Abstract

We study $K$-stability for tensor products of diagonal AH-algebras with arbitrary C*-algebras. Our main result provides a characterization of $K$-stability: for a diagonal AH-algebra $A = \varinjlim (A_i, \varphi_i)$, $A \otimes B$ is $K$-stable for every C*-algebra $B$ if and only if the sizes of the matrix blocks in the inductive system grow without bound. As applications, we show that non-$\mathcal{Z}$-stable Villadsen algebras of the first kind are $K$-stable when tensored with any C*-algebra. Moreover, any simple, unital, infinite-dimensional diagonal AH-algebra automatically satisfies this growth condition, and therefore its tensor product with arbitrary C*-algebras is always $K$-stable.