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

TL;DR

The article addresses when tensoring a diagonal AH‑algebra $A$ with an arbitrary C*-algebra $B$ preserves $K$‑stability. It proves a sharp criterion: $A\otimes B$ is $K$‑stable for all $B$ precisely when the minimal matrix size in the homogeneous summands of the inductive system defining $A$ grows without bound, i.e., $\lim_i d_{A_i}=\infty$. This growth condition is shown to be equivalent to tensorial $K$‑stability, enabling applications to non‑$\mathcal{Z}$‑stable Villadsen algebras and to simple, unital, infinite‑dimensional diagonal AH‑algebras, whose tensor products with any $B$ are then $K$‑stable. The results connect nonstable $K$‑theory with standard $K$‑theory in a broad class and explain when tensor products preserve topological information captured by unitary homotopy groups.

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.

Theorems & Definitions (21)

• Theorem A: Theorem \ref{['mainthm']}
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Definition 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3