The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability

Proves a separable C*-algebra absorbs the Jiang–Su algebra exactly when its central-sequence algebra has no characters.

1 paper September 25, 2026 Explained by AI

The Jiang–Su algebra Z is a small building block; an algebra that absorbs it, so that A ⊗ Z looks like A, is well behaved. A character is a map to ordinary numbers that respects sums and products. Kirchberg and Rørdam asked whether absorbing Z is the same as a related algebra having no characters. This paper proves it is, for every unital separable C*-algebra, and Lean checks it. It also shows infinite tensor powers of characterless algebras absorb Z.

First step

2015

Kirchberg and Rørdam, Question 3.1 in "When central sequence C*-algebras have characters"

Significance

Major

It answers a question of Kirchberg and Rørdam, listed in a recent problem collection, for all unital separable C*-algebras, with a Lean check.

Trust

Formal proof listed

Field

Operator algebras

1 paper

The explanation

Start with 2-by-2 matrices, M₂. A character is a map χ to ordinary numbers that keeps sums and products. Write e11, e12, e21, e22 for the matrix units. Since e11·e11 = e11, χ(e11) is 0 or 1. Since e12·e21 = e11 and e21·e12 = e22, χ(e11) = χ(e22). But e11 + e22 is the identity, so χ(e11) = 1/2. That is a contradiction, so M₂ has no character. The same holds for any algebra built only from matrix blocks of size 2 or more.

A C*-algebra is a family of operators closed under sums, products, adjoints and limits. The Jiang–Su algebra Z is a special C*-algebra with no characters that behaves like ordinary numbers in many ways. An algebra A is Z-stable when A ⊗ Z is again ASource 1. Z-stability is a key regularity property in classifying these algebras. The central-sequence algebra F_ω(A) collects sequences in A that asymptotically commute with all of A.

Kirchberg and Rørdam proved A is Z-stable exactly when F_ω(A) holds a suitable characterless subalgebraSource 2. Then F_ω(A) has no characters at all. They asked whether the converse holds: does no characters on F_ω(A) already give Z-stabilitySource 3? The question also appears as Problem LXXX in a recent problem collectionSource 4. The obstacle is to build a characterless piece of bounded matrix size from nothing but the absence of characters.

The paper proves the converse, for every nonzero unital separable complex C*-algebra and every free ultrafilterSource 5. No nuclearity, simplicity or trace hypothesis is neededSource 6. The key step maps a dimension-drop algebra I(2,3) into a finite tensor power of any characterless algebraSource 7. I(2,3) is a family of 6-by-6 matrices along [0, 1]. At the ends they shrink to blocks of size 2 and 3. Its irreducible pieces have size 2, 3 and 6, so by the M₂ argument it has no characterSource 8. Z itself is built from pieces of this kind, whose two block sizes share no factor. As a corollary, the infinite minimal tensor power of any characterless separable algebra is Z-stableSource 9.

How sure is this? Lean, a computer proof checker, has a formal statement of the main criterion: F_ω(A) has no character exactly when A is Z-stable. So the paper proves this, as far as the Lean statement matches the paper. The tensor-power corollary is not named in the Lean scope note. No specialist review is published.

What it doesn't show. This does not show that a finite tensor power of a characterless algebra absorbs Z; the absorption result is for the infinite power.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

A clean answer to a named question, with a Lean check of the main criterion. The tensor-power corollary rests on the paper alone. No specialist review is published.

Words used

C*-algebra: A family of operators on a Hilbert space closed under sums, products, adjoints and limits.
Character: A nonzero map from the algebra to the complex numbers that keeps sums, products and adjoints.
Jiang–Su algebra Z: A special infinite-dimensional C*-algebra with no characters that acts like the number 1 under tensor products.
Z-stable: An algebra A with A ⊗ Z isomorphic to A; a regularity property used in classification.
Central-sequence algebra F_ω(A): Bounded sequences in A that asymptotically commute with every element of A, taken along an ultrafilter.
Free ultrafilter: A rule for taking limits of every bounded sequence that ignores any finite set of terms.
Dimension-drop algebra I(2,3): Continuous paths of 6-by-6 matrices on [0, 1] that drop to blocks of size 2 at 0 and 3 at 1.
Tensor power: The algebra D ⊗ D ⊗ … ⊗ D built from several copies of D.

The original paper

The Kirchberg–Rørdam character criterion (PDF, opens in a new tab)
September 25, 2026 Formal proof listed

Lean notes

All quoted lines (9)

An algebra is \emph $\Z$ -stable if it is isomorphic to its minimal tensor product with $\Z$ .
The Kirchberg–Rørdam character criterion Back to text
In the case of $\Z$ , Kirchberg and R\o rdam proved that a unital separable algebra is $\Z$ -stable exactly when its norm central-sequence algebra contains, with the same unit, a separable subhomogeneous algebra without characters.
The Kirchberg–Rørdam character criterion Back to text
Kirchberg and R\o rdam asked whether the absence of characters on $F_\omega(A)$ characterizes Jiang–Su absorption among unital separable $C^*$ -algebras.
The Kirchberg–Rørdam character criterion Back to text
The character question also appears as Problem LXXX in the recent collection of Schafhauser, Tikuisis, and White.
The Kirchberg–Rørdam character criterion Back to text
We prove the Kirchberg–R\o rdam character criterion: for every free ultrafilter on $\N$ , a nonzero unital separable complex $C^*$ -algebra absorbs the Jiang–Su algebra if and only if its norm central-sequence algebra has no characters.
The Kirchberg–Rørdam character criterion Back to text
The theorem applies to all such $A$ , with no nuclearity, simplicity, trace, dimension, or comparison assumption.
The Kirchberg–Rørdam character criterion Back to text
There is an integer $m\ge1$ and a unital $*$ -homomorphism $I(2,3)\longrightarrow D^{\otimes_{\max}m},$ where $I(2,3)=\{f\in C([0,1],M_2\otimes M_3): f(0)\in M_2\otimes1_3,\ f(1)\in1_2\otimes M_3\}.$
The Kirchberg–Rørdam character criterion Back to text
The resulting irreducible dimensions are $2$ at $0$ , $3$ at $1$ , and $6$ in the interior.
The Kirchberg–Rørdam character criterion Back to text
We also prove that the infinite minimal tensor power of every nonzero unital separable complex $C^*$ -algebra without characters is $\Z$ -stable and contains a unital copy of $\Z$ .
The Kirchberg–Rørdam character criterion Back to text

Other sources

openai/math, Lean scope note for family 299 No nuclearity, simplicity, trace, or comparison hypothesis is imposed.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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