Does strict comparison by traces force pureness in simple C*-algebras?

Strict comparison by traces forces pureness

The paper's central answer is yes for unital, simple, nonelementary C*-algebras: strict comparison by traces implies the algebra is pure. Pureness means the Cuntz semigroup Cu(A) is both almost unperforated and almost divisible. Strict comparison by traces already delivers almost unperforation, so the real work is producing almost divisibility.

The mechanism is a new uniform density result for ranks of operators. For a positive element a in the stabilization A⊗K, its rank at a lower semicontinuous trace τ is dτ(a) = lim τ(a^(1/n)). Theorem A shows that for any strictly positive continuous affine function f on the tracial state space T1(A) and any ε > 0, there is m ≥ 1 and e in Mm(A)+ with |f(τ) − dτ(e)| ≤ ε for every τ in T1(A). In other words, ranks of operators are uniformly dense in the strictly positive affine functions on the tracial state space.

That density is what converts a comparison hypothesis into divisibility. The proof needs no separability, exactness, stable rank one, or regularity of the extreme boundary, so the conclusion is not an artifact of those extra hypotheses.

Ranks are dense, not exactly realized

A reader could easily conflate two different questions. Exact rank realization asks whether a given function is literally the rank of some element. Uniform density asks only whether ranks come arbitrarily close, uniformly over all traces. Theorem A settles the second, weaker question, and that is enough for the pureness application.

The distinction matters because earlier exact realization results required additional regularity, such as hypotheses on the trace boundary or on ranks already available in the algebra. The present approximation theorem drops the stable rank hypothesis but works with ordinary traces rather than 2-quasitraces. The paper explicitly separates strict comparison by ordinary traces from strict comparison by 2-quasitraces, the latter being the usual unqualified comparison property.

Two lemmas carry the approximation. Lemma 2.1 compresses a positive element into the hereditary subalgebra Her(b) of a full element b while preserving τ(c) = τ(a) for all tracial states. Lemma 2.2 converts an evaluation into a rank function: for c in A+ and integer m ≥ max{1, ‖c‖}, the element e = ⊕_{j=1}^m (c − j)+ in Mm(A)+ satisfies dτ(e) ≤ τ(c) ≤ dτ(e) + dτ(c) for every lower semicontinuous trace τ. The error is bounded by the rank of c, independently of its norm.

Toms–Winter holds for approximately subhomogeneous algebras

Combining Corollary B with known absorption and dimension theorems yields an equivalence for separable, unital, simple, nonelementary C*-algebras with locally finite nuclear dimension: finite nuclear dimension, Z-stability, strict comparison, and pureness are all equivalent, and in that case dim_nuc(A) ≤ 1.

Because approximately subhomogeneous C*-algebras have locally finite decomposition rank and hence locally finite nuclear dimension, the Toms–Winter conjecture holds for separable, unital, simple, nonelementary ASH algebras. This is an application of the pureness theorem, not a separate approximation argument.

The equivalence is stated for algebras with locally finite nuclear dimension. It is not asserted for arbitrary simple nonelementary C*-algebras, where the pureness conclusion still holds under strict comparison by traces but the Z-stability and finite-nuclear-dimension equivalences are not claimed by this theorem.

The Global Glimm Property makes ranks dense and divisible

The general results rest on the Global Glimm Property. The paper first treats the rank problem abstractly for Cu-semigroups: for a Cu-semigroup S and compact K ⊆ F(S), density of ranks on K is equivalent to almost divisibility of ranks on K. A sufficient criterion is given in terms of a weak form of (2, ω)-divisibility together with a smoothing property.

Specializing to Cuntz semigroups, the paper shows that Cu(A) always satisfies the required smoothing property with respect to the cone T(A) of lower semicontinuous traces. Consequently, three conditions are equivalent: ranks are dense on traces, ranks are almost divisible on traces, and A has the tracial Global Glimm Property.

Corollary 4.3 then states that if A has the Global Glimm Property, ranks are dense and almost divisible on traces. The Global Glimm Property is equivalent to (2, ω)-divisibility of Cu(A), and also to (n, ω)-divisibility for every n ≥ 2, and it passes to stabilization.

Pure exact algebras are exactly those with both properties

Theorem 4.4 states that every C*-algebra with the Global Glimm Property and strict comparison by traces is pure. Strict comparison by traces gives almost unperforation; the Global Glimm Property plus Corollary 4.3 gives almost divisibility of ranks on traces, which strict comparison upgrades to almost divisibility in the Cuntz semigroup.

Corollary 4.5 turns this into a characterization for exact algebras: an exact C*-algebra is pure if and only if it has strict comparison and the Global Glimm Property. The forward direction holds in general, since every pure C*-algebra has strict comparison and the Global Glimm Property. The converse uses that for exact algebras every lower semicontinuous 2-quasitrace is a trace, so strict comparison agrees with strict comparison by traces.

This is the paper's sharpest general statement. It replaces a one-directional implication with a two-sided criterion, but only under exactness, which is what makes the two comparison notions coincide.

Where the conclusions apply and where they stop

The pureness conclusion of Corollary B is stated for unital, simple, nonelementary C*-algebras with strict comparison by traces. The proof of Theorem A uses simplicity and nonelementarity only to obtain full elements of arbitrarily small rank uniformly on T1(A); the paper notes the conclusion therefore holds for every unital C*-algebra with that property.

The Toms–Winter equivalence is narrower: it requires separability, unitality, simplicity, nonelementarity, and locally finite nuclear dimension. The exact-algebra characterization requires exactness. These are different scopes, and the paper keeps them separate rather than presenting one as a general theorem.

The paper also distinguishes strict comparison by traces from strict comparison by 2-quasitraces throughout. For exact algebras the two agree, which is precisely why the exact case admits the if-and-only-if characterization while the general case does not.

The decisive variable is which comparison and which property

The most useful way to read this paper is as a map of which combination of hypotheses yields pureness. For unital, simple, nonelementary C*-algebras, strict comparison by traces alone suffices. For general C*-algebras, the Global Glimm Property plus strict comparison by traces suffices. For exact C*-algebras, those two conditions are also necessary, giving an exact characterization.

The rank density theorem is the engine, and its strength is uniformity over all traces with no stable rank hypothesis. The trade-off is that it delivers approximation, not exact realization, and it concerns ordinary traces rather than 2-quasitraces. Readers comparing this with earlier exact realization results should track that difference rather than treating the results as interchangeable.

The paper's own priority is the equivalence structure: strict comparison by traces, pureness, Z-stability, and finite nuclear dimension coincide under locally finite nuclear dimension, and the Toms–Winter conjecture follows for approximately subhomogeneous C*-algebras. The decisive question for any given algebra is therefore not whether strict comparison holds in the abstract, but which comparison property it has and whether the Global Glimm Property or exactness is available.

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