No scalar shadows in central sequences: a criterion for Jiang–Su stability

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Family 299 · Operator algebras
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

An operator algebra can contain sequences that increasingly commute with every fixed element. These central sequences reveal internal structure that may be invisible in the algebra’s ordinary center.

The Kirchberg–Rørdam character question asks whether one very weak-looking condition on this sequence algebra already forces a strong regularity property: if central sequences admit no one-dimensional representation, must the original algebra absorb the Jiang–Su algebra?

Family 299 claims yes for every nonzero unital separable complex C*-algebra, without nuclearity, simplicity or trace assumptions. It also claims that the infinite spatial tensor power of every such characterless algebra absorbs, and contains a unital copy of, the Jiang–Su algebra. The main ingredient is a finite maximal-tensor construction of a dimension-drop algebra image.

2. Background and essential definitions

A C*-algebra is a norm-complete algebra closed under adjoints, with $\|a^*a\|=\|a\|^2$. A character is a nonzero *-homomorphism from the algebra to $\mathbb C$. For a unital algebra it automatically preserves the identity. Characters are one-dimensional representations, so having none means that no nonzero representation reduces all algebraic behavior to scalars.

For a free ultrafilter $\omega$ on $\mathbb N$, the norm ultrapower is

$$A_\omega=\ell^\infty(\mathbb N,A)/ \{(a_n):\lim_\omega\|a_n\|=0\}.$$

Bounded sequences are identified when their difference tends to zero in operator norm along the ultrafilter. The norm central-sequence algebra is

$$F_\omega(A)=A_\omega\cap A',$$

where $A$ sits as constant sequences. Its elements have representatives asymptotically commuting with every fixed element of $A$. This is not a tracial ultrapower, which would discard sequences small in a trace-based quadratic norm.

The Jiang–Su algebra $\mathcal Z$ is a distinguished infinite-dimensional algebra whose absorption expresses regularity:

$$A\cong A\otimes_{\min}\mathcal Z.$$

The minimal tensor product is the spatial completion obtained from Hilbert-space representations. The maximal tensor product uses the largest C*-norm allowed by representations of the algebraic tensor product. They need not coincide for the nonnuclear algebras allowed here.

A subhomogeneous algebra has a common finite upper bound on dimensions of its irreducible representations. A full positive element generates the entire algebra as a closed two-sided ideal. Fullness says that the element is visible in every nonzero irreducible representation, not that it is invertible.

3. Development history and previous partial results

Toms–Winter’s theory of strongly self-absorbing algebras formalized how central copies of a fixed algebra imply tensorial absorption. For $\mathcal Z$, Kirchberg–Rørdam proved a useful criterion: a unital separable algebra absorbs $\mathcal Z$ exactly when its norm central-sequence algebra contains a same-unit separable subhomogeneous algebra without characters.

The necessary direction is immediate at the level of characters: if the central-sequence algebra had a character, restricting it to that unital subalgebra would give a forbidden character there. The open question was whether the absence of characters on the whole central-sequence algebra suffices to construct the required subalgebra.

Dadarlat–Toms related absorption to infinite tensor powers. They showed that a suitable characterless subhomogeneous subalgebra inside such a tensor power gives Jiang–Su stability. Their dimension-drop constructions covered recursive subhomogeneous algebras whose matrix sizes are sufficiently large relative to topological dimension.

Other regularity criteria involved stronger hypotheses. Robert–Tikuisis used full orthogonal central positive elements under finite nuclear dimension. Perera–Thiel–Vilalta characterized absorption through comparison and divisibility properties of the central-sequence Cuntz semigroup. M1 seeks a criterion requiring none of those comparison or dimension conditions.

The companion dimension manuscript in family 291 constructs two full orthogonal positive elements in a finite maximal tensor power of a characterless algebra. M1 describes a different representation-label construction and then carries the process further, to a unital dimension-drop map. It also uses the companion’s commuting-copy lemma, with the earlier reindexing provenance credited to Kirchberg–Rørdam.

4. What the new manuscripts claim

For every nonzero unital separable complex C*-algebra $A$ and every free ultrafilter $\omega$ on $\mathbb N$, M1 states

$$F_\omega(A)\text{ has no characters} \quad\Longleftrightarrow\quad A\cong A\otimes_{\min}\mathcal Z.$$

There is no nuclearity, simplicity, trace, dimension or comparison assumption. The condition is imposed on $F_\omega(A)$, not merely on $A$ itself. The statement holds for each free ultrafilter, so the resulting absorption property does not depend on choosing a favorable one.

The finite construction applies more broadly. For any nonzero unital complex C*-algebra $D$ with no characters, even nonseparable, M1 claims that some finite $m\geq1$ admits a unital *-homomorphism

$$I(2,3)\longrightarrow D^{\otimes_{\max}m}.$$

The prime dimension-drop algebra here is

$$\begin{aligned} I(2,3)=\{f\in C([0,1],M_2\otimes M_3): &\ f(0)\in M_2\otimes1_3,\\ &\ f(1)\in1_2\otimes M_3\}. \end{aligned}$$

Its interior fibers are six-by-six matrices, while endpoint restrictions have sizes two and three. These dimensions explain the absence of one-dimensional representations and its role as a building block for $\mathcal Z$.

The map need not be injective. Its image is nevertheless a unital subhomogeneous algebra without characters, which is the property required by the absorption criteria. The finite exponent depends on $D$, and M1 does not claim that this finite tensor power itself absorbs $\mathcal Z$.

For nonzero unital separable characterless $D$, the infinite minimal tensor power

$$B=D^{\otimes_{\min}\infty} =\varinjlim\bigl(D^{\otimes_{\min}n},x\mapsto x\otimes1\bigr)$$

is claimed to satisfy $B\cong B\otimes_{\min}\mathcal Z$ and to contain a unital copy of $\mathcal Z$. The finite theorem uses maximal products; this infinite conclusion uses minimal products. They are deliberately different statements.

5. Why the advance matters

Importance: 5/5. If correct, the theorem characterizes a major regularity property by an elementary representation-theoretic obstruction in central sequences. It would replace substantial comparison or approximation assumptions with the absence of scalar representations.

Its scope extends beyond the nuclear classification setting. Nonnuclear, nonsimple algebras and algebras with complicated traces are included, provided they are nonzero, unital and separable. That breadth is part of what makes the criterion striking.

The infinite-tensor conclusion also explains how repeated composition can create regularity. A single characterless algebra need not already absorb $\mathcal Z$, but infinitely many spatial tensor copies would supply enough distributed structure. This should not be compressed into the false stronger slogan that characterlessness of $A$ alone implies its absorption.

6. The main idea

Detect noncommutativity in every representation

The absence of characters means that every irreducible representation has dimension at least two. M1 first extracts finite-dimensional spaces of self-adjoint elements whose images detect this noncommutativity.

A tensor-label construction keeps selected vectors linearly independent after arbitrary rank-at-least-two maps on individual tensor factors. These robust labels produce self-adjoint elements that fail to commute in every irreducible representation of a suitable finite tensor power.

A double commutator then yields full positive and negative parts. A further labeling step combines finitely many connections between them into one full square-zero element. “Square zero” means $v^2=0$, allowing its source and range pieces to behave like off-diagonal matrix coordinates.

Go beyond two full orthogonal pieces

The square-zero element supplies a homomorphism from a matrix cone, equivalently an orthogonality-preserving completely positive map on a matrix algebra. Its first corner is full, but fullness alone does not give a unital dimension-drop map.

The next stage arranges two exactly equivalent orthogonal positive contractions, represented by $v^*v$ and $vv^*$, and controls the part of the identity they leave unused. A simplex of matrix embeddings organizes the construction over a finite tensor power. Regions near coordinate axes supply room to absorb the remaining defect after a further finite amplification.

This is the comparison input needed for the Rørdam–Winter dimension-drop criterion. It upgrades full orthogonal elements to the specific unital $I(2,3)$ image, without assuming stable rank one.

Place the finite construction centrally

Inside a central-sequence algebra, reindexing supplies commuting copies of a suitable separable characterless subalgebra. Commuting copies define a map from a maximal tensor product, so the finite theorem produces the required dimension-drop image centrally. Kirchberg–Rørdam’s established criterion then gives $\mathcal Z$-absorption of $A$.

For the infinite minimal tensor power, disjoint tensor blocks provide the needed copies. The finite image and the Dadarlat–Toms absorption theorem give the infinite-power conclusion. Unitality is preserved through these passages, which matters because a zero or nonunital map would not supply the stated criterion.

7. Scope, limitations, and open questions

The main theorem is for nonzero unital separable complex C*-algebras. It does not simply assert the same formula for nonunital algebras, where central-sequence annihilator quotients require additional care. Its ultrapower is defined with the operator norm, not a chosen collection of traces.

The auxiliary finite theorem drops separability but uses maximal tensor products. Its conclusion is a homomorphism with a useful image, not necessarily an embedding of $I(2,3)$, and not finite-power Jiang–Su absorption. No uniform exponent is given independently of $D$.

The infinite-power theorem returns to separability and minimal tensor products. It asserts both absorption and an actual unital copy of $\mathcal Z$ at that infinite stage. Whether stronger finite-stage conclusions hold under additional assumptions is a different question.

Finally, M1’s dimension-drop comparison and its reindexing dependencies are essential parts of the claimed construction. Merely finding two full orthogonal elements would not, without further hypotheses or steps, reproduce the entire theorem.

8. Sources and further reading

• Kirchberg and Rørdam, When central sequence C*-algebras have characters, for the character criterion question and its connection to tensor powers.
• Kirchberg and Rørdam, Central sequence C*-algebras and tensorial absorption of the Jiang–Su algebra, for the earlier central-sequence absorption framework.
• M1 states the precise Dadarlat–Toms and Rørdam–Winter criteria used to pass from its finite construction to absorption.

Source manuscripts

• M1. OpenAI, The Kirchberg–Rørdam character criterion (September 25, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.