pith. sign in

arxiv: 2512.04780 · v2 · pith:4MQ5HY3Qnew · submitted 2025-12-04 · 🧮 math.OA

Tensorial Permanence of K-Stability for Diagonal AH-Algebras

Apurva Seth This is my paper

Pith reviewed 2026-05-25 07:55 UTC · model grok-4.3

classification 🧮 math.OA
keywords diagonal AH-algebrasK-stabilitytensor productsinductive limitsmatrix block sizesVilladsen algebrasC*-algebraspermanence properties
0
0 comments X

The pith

A diagonal AH-algebra tensored with any C*-algebra is K-stable precisely when matrix block sizes in its inductive system grow without bound.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes K-stability under arbitrary tensor products for diagonal AH-algebras presented as inductive limits. It proves that A ⊗ B remains K-stable for every C*-algebra B if and only if the matrix block sizes appearing in the approximating algebras increase without bound. This supplies an explicit, checkable criterion for a permanence property of K-theory. The characterization immediately yields that non-Z-stable Villadsen algebras of the first kind satisfy the property and that every simple unital infinite-dimensional diagonal AH-algebra does as well.

Core claim

For a diagonal AH-algebra A equal to the inductive limit of a system of diagonal AH-algebras with diagonal-preserving connecting maps, A ⊗ B is K-stable for every C*-algebra B if and only if the sizes of the matrix blocks in the approximating algebras grow without bound.

What carries the argument

The growth condition on matrix block sizes in the inductive system of diagonal AH-algebras.

If this is right

  • Non-Z-stable Villadsen algebras of the first kind are K-stable when tensored with any C*-algebra.
  • Every simple unital infinite-dimensional diagonal AH-algebra has K-stable tensor products with arbitrary C*-algebras.
  • The tensorial permanence of K-stability is decided exactly by whether matrix block sizes become arbitrarily large.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar growth conditions on approximations might characterize K-stability for non-diagonal AH-algebras.
  • The separation between K-stability and Z-stability shown by the Villadsen examples suggests these two permanence properties are independent in general.
  • A construction of a bounded-size diagonal AH-algebra together with a specific B that breaks K-stability would give an explicit counterexample to the property.

Load-bearing premise

The inductive system consists of diagonal AH-algebras whose connecting maps preserve the diagonal structure, and K-stability is the standard notion from the surrounding C*-algebra literature.

What would settle it

An explicit diagonal AH-algebra whose matrix block sizes remain bounded, together with a concrete C*-algebra B such that A ⊗ B fails to be K-stable.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies K-stability of tensor products involving diagonal AH-algebras. Its central claim is an if-and-only-if characterization: for a diagonal AH-algebra A = lim (A_i, φ_i), the tensor product A ⊗ B is K-stable for every C*-algebra B if and only if the matrix block sizes appearing in the inductive system grow without bound. Applications include that non-Z-stable Villadsen algebras of the first kind remain K-stable after tensoring with arbitrary C*-algebras, and that every simple unital infinite-dimensional diagonal AH-algebra automatically satisfies the growth condition and hence has K-stable tensor products with all C*-algebras.

Significance. If the stated equivalence holds, the result supplies a concrete, checkable criterion for tensorial permanence of K-stability within the class of diagonal AH-algebras. The applications to Villadsen examples and to the automatic case for simple infinite-dimensional algebras are useful for the structure theory of C*-algebras and their K-theoretic invariants. The if-and-only-if form against an external definition of K-stability is a strength.

minor comments (2)
  1. The abstract and introduction would benefit from an explicit pointer to the precise definition of K-stability used (presumably in §2) so that readers can immediately locate the external reference against which the if-and-only-if is proved.
  2. Notation for the inductive limit (varinjlim) and the connecting maps φ_i is standard but could be accompanied by a brief reminder of the diagonal structure preservation assumption on the maps, even if it is implicit in the class of diagonal AH-algebras.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report. The manuscript has no major comments requiring response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states an if-and-only-if characterization of tensorial K-stability for diagonal AH-algebras in terms of unbounded matrix-block growth in the inductive system. This equivalence is presented against external definitions of K-stability (section 2) and standard facts about AH-algebra simplicity and Villadsen examples. No load-bearing self-citations, self-definitional steps, fitted inputs renamed as predictions, or ansatzes smuggled via prior work appear in the abstract or described claims. The central result does not reduce to its inputs by construction and remains independent of the paper's own fitted quantities or uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard domain assumptions in C*-algebra theory; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Standard definitions of diagonal AH-algebras, inductive limits, and K-stability from the surrounding literature.
    The characterization is stated relative to these pre-existing notions.

pith-pipeline@v0.9.0 · 5664 in / 1154 out tokens · 37204 ms · 2026-05-25T07:55:32.234044+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    A class of simple C*-algebras with stable rank one

    [EHT09] George A Elliott, Toan M Ho, and Andrew S Toms. “A class of simple C*-algebras with stable rank one”. In: Journal of Functional Analysis 256.2 (2009), pp. 307–322. [EN25] George A Elliott and Zhuang Niu. “On the small boundary property and Z-absorption, II”. In: arXiv preprint arXiv:2504.03611 (2025). [Han78] David Handelman. “ K0 of von Neumann a...

    work page arXiv 2009

  2. [2]

    The homotopy groups of the unitary groups of non-commutative tori

    [Rie87] Marc A Rieffel. “The homotopy groups of the unitary groups of non-commutative tori”. In: Journal of Operator Theory (1987), pp. 237–254. [RLL00] Mikael Rørdam, Flemming Larsen, and Niels Laustsen. An introduction to K-theory for C*-algebras. Vol

    work page 1987

  3. [3]

    AF-algebras and rational homotopy theory

    [SV20a] Apurva Seth and Prahlad Vaidyanathan. “AF-algebras and rational homotopy theory”. In: New York Journal of Mathematics 26 (2020), pp. 931–949. [SV20b] Apurva Seth and Prahlad Vaidyanathan. “K-Stability of Continuous C(X)-Algebras”. In: Proceedings of the American Mathematical Society 148.9 (2020), pp. 3897–3909. [SV23] Apurva Seth and Prahlad Vaidy...

    work page 2020