A quasitrace that cannot be a trace—and finite algebras with an infinite tensor product

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Family 294 · Operator algebras
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

The trace of a matrix measures size additively: $\operatorname{Tr}(a+b)=\operatorname{Tr}(a)+\operatorname{Tr}(b)$. It also assigns the same value to $x^*x$ and $xx^*$. In more general operator algebras, a quasitrace retains positivity and this symmetry, but initially requires additivity only for commuting operators.

Does compatibility with two-by-two matrices force full additivity? Kaplansky’s quasitrace problem asks whether these weaker size functions must actually be traces. Family 294 claims a negative answer, with a fixed pair of positive operators witnessing a definite additive defect for every normalized 2-quasitrace on the example.

An established chain of implications then yields another striking conclusion: two simple stably finite C*-algebras can have a properly infinite spatial tensor product. The paper distinguishes several different algebras used for these conclusions; no single example is asserted to have every listed property.

2. Background and essential definitions

A C*-algebra is a norm-complete adjoint algebra satisfying $\|x^*x\|=\|x\|^2$. A tracial state is a positive linear functional $\tau$ with $\tau(1)=1$ and $\tau(xy)=\tau(yx)$. It is a normalized analogue of matrix trace.

A quasitrace is positive on positive elements, respects $\tau(x^*x)=\tau(xx^*)$, is linear on commutative subalgebras, and is compatible with real and imaginary parts. A 2-quasitrace additionally extends consistently to $M_2(A)$. Its extension takes value two on the two-by-two identity when the original quasitrace takes value one on $1_A$. This is an unnormalized matrix extension.

An isometry satisfies $v^*v=1$. It is unitary if also $vv^*=1$. An algebra is finite if every isometry is unitary, and stably finite if this holds in every finite matrix algebra over it. It is properly infinite if it contains two isometries with orthogonal ranges, so that two full copies of its unit fit inside the unit.

“No matrix algebra is properly infinite” is weaker than stable finiteness. It excludes two orthogonal full copies at every matrix size, but does not by definition exclude every individual nonunitary isometry. This distinction matters for the constructed algebra.

An AW-algebra* has an intrinsic projection structure modeled on von Neumann algebras. Kaplansky asked whether every type $\mathrm{II}_1$ AW*-factor is a von Neumann algebra. Here that means an abstract W*-algebra admitting a Banach-space predual, not merely an algebra given some representation on a Hilbert space. The finite-factor question is equivalent to the general 2-quasitrace problem used by the manuscript.

3. Development history and previous partial results

Kaplansky introduced AW*-algebras in 1951 to describe operator-algebra structure without choosing a Hilbert-space representation. The factor types behave differently: type I AW*-factors are W*-algebras, while type III counterexamples exist. The type $\mathrm{II}_1$ question isolates the finite case.

Cuntz’s dimension functions and the work of Blackadar–Handelman connected quasitraces with size and comparison of positive operators. A key existence theorem says that a unital algebra admits a normalized 2-quasitrace if no matrix algebra over it is properly infinite. This offers a route to a counterexample: arrange that obstruction to proper infiniteness while separately ruling out tracial states.

Haagerup proved that 2-quasitraces on unital exact C*-algebras are traces. Exactness is a tensor-product preservation property satisfied by nuclear algebras and many other examples. His argument circulated from 1991 and was published in 2014. Therefore any counterexample to full additivity must lie outside exactness.

The matrix-extension requirement cannot be skipped. A previously reported example of a 1-quasitrace failing to be a 2-quasitrace already shows that weaker quasitraces need not be linear, but does not answer the present question.

Milhøj–Rørdam connected the quasitrace problem to whether spatial tensor products preserve stable finiteness for simple unital algebras. They also described how a negative quasitrace answer would yield a properly infinite tensor product with a reduced free-group algebra. M1 constructs the missing example and states that the two-generator free group suffices.

4. What the new manuscripts claim

M1 constructs a separable unital complex C*-algebra $A$ admitting a normalized 2-quasitrace and positive contractions $a,b\in A$ such that every normalized 2-quasitrace satisfies

$$\tau(a+b)-\tau(a)-\tau(b)\geq\frac1{144}.$$

A positive contraction is a positive operator of norm at most one. The inequality makes nonadditivity uniform over the quasitraces, not merely a feature of one specially chosen functional. In particular, $A$ has no tracial state. By Haagerup’s theorem it is necessarily nonexact.

The construction begins with another separable unital algebra

$$B=C^*(1,x_1,\ldots,x_{24})$$

satisfying

$$\sum_{i=1}^{24}x_i^*x_i=1, \qquad \sum_{i=1}^{24}x_ix_i^*\leq\frac23,1,$$

while no $M_m(B)$ is properly infinite. These relations rule out a tracial state: additivity and the trace symmetry would assign equal values to the two sums, contrary to the bounds. The main quantitative example is obtained in $A=M_{48}(B)$.

M1 then derives three related consequences:

• Two unital simple stably finite complex C*-algebras can have a properly infinite minimal tensor product. One factor can be $C_r^*(\mathbb F_2)$, the reduced algebra of the free group on two generators.
• The separable algebra $B$ admits a normalized 2-quasitrace, but $B\otimes_{\min}C_r^*(\mathbb F_2)$ is properly infinite and admits none.
• A separable unital stably finite complex C*-algebra can have no tracial state.

The first uses a type $\mathrm{II}_1$ AW*-factor with no asserted separability. The second uses the explicit $B$, for which stable finiteness is not claimed. The third uses a finitely generated subalgebra of the AW*-factor, with no simplicity assertion. These qualifications prevent an unjustified stronger combined theorem.

5. Why the advance matters

Importance: 5/5. If correct, the construction separates quasitracial size from linear trace in the general C*-algebra setting and answers Kaplansky’s finite AW*-factor question negatively. The explicit positive pair makes the failure quantitative.

The tensor consequence would show that stable finiteness is not preserved by combining arbitrary simple unital algebras through the spatial tensor product. One factor may be the familiar reduced free-group algebra, so the phenomenon does not require both factors to be newly invented examples.

The result leaves Haagerup’s exact-algebra theorem intact. Rather than making quasitraces unreliable everywhere, it identifies behavior that exactness prevents. It also highlights why several definitions containing the word “finite” must be tracked separately.

6. The main idea

Preserve a trace obstruction with finitely many generators

The two displayed sums provide a compact certificate that no tracial state exists. Keeping the generator count fixed is important: the same finite obstruction must survive while infinitely many possible matrix-isometry constructions are excluded.

M1 realizes the generators as shifts between finite-rank vector bundles. A vector bundle is a family of finite-dimensional vector spaces varying continuously over a compact base. Nontrivial topology can prevent a prescribed trivial bundle from fitting into another bundle, even when a naive pointwise rank comparison looks plausible.

Test every matrix size and every polynomial complexity

For each pair $(m,R)$, the construction chooses a compact base and bundles designed to obstruct a candidate pair of isometries in $M_m(B)$ expressed by words of length at most $R$.

Such a polynomial matrix restricts to a map from a trivial bundle into a finite sum of the chosen bundles. If it approximated the required isometries sufficiently well, its image would force a forbidden trivial subbundle. A nonzero Chern class, a topological invariant measuring twisting, obstructs that inclusion.

All these finite tests are assembled in one bounded operator product. The generated C*-algebra remains separable because it has finitely many generators. Any hypothetical pair of isometries can be approximated by finite words; its matrix size and word length select a test already included in the product.

Normalize without destroying the topology

The bundle maps must simultaneously satisfy the column-sum identity and strict row-sum bound. M1 uses an operator-scaling construction based on a convex determinant potential. A unique minimizing normalization varies continuously and respects changes of bundle coordinates.

Different collections of maps handle small and large source subspaces, keeping the total at 24 as the topological tests grow. This uniform finite length preserves the trace obstruction through the product construction.

Extract a definite nonlinear defect

The absence of proper infiniteness in every matrix size gives normalized 2-quasitraces by the established existence theorem. Passing to $M_{48}(B)$, a finite telescoping argument uses linearity only where the relevant elements commute. It produces the fixed pair $a,b$ and the lower bound $1/144$, without requiring an explicit formula for any chosen quasitrace.

The tensor-product and AW*-factor conclusions then use established implications from the quasitrace problem. Their examples inherit only the properties explicitly retained by those deductions.

7. Scope, limitations, and open questions

All tensor products in the stated consequences are minimal, or spatial, C*-tensor products. The paper does not replace them by arbitrary tensor norms. Its nonexact examples do not contradict trace additivity for exact algebras.

The algebra $B$ is known in the manuscript to have no properly infinite matrix amplification, which suffices for quasitrace existence. Calling it stably finite would add an unproved property. Similarly, the simple AW*-factor used in the tensor example is not asserted separable, and the separable stably finite traceless algebra is not asserted simple.

The main theorem gives a uniform additive gap but does not classify all nonlinear 2-quasitraces or provide a simple closed formula for them. Understanding more explicit quasitrace values and determining which further structural assumptions restore tensor-product finiteness are questions beyond the stated conclusions.

8. Sources and further reading

• Haagerup, Quasitraces on exact C*-algebras are traces, for the exactness boundary and the historical equivalences.
• Milhøj and Rørdam, Around traces and quasitraces, for the tensor-product question and its connection to quasitraces.
• M1 distinguishes the three algebras in its consequences and identifies the characteristic-class and scaling ingredients of its construction.

Source manuscripts

• M1. OpenAI, A counterexample to Kaplansky’s quasitrace conjecture and failure of tensor-product stable finiteness (September 23, 2026).

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