Nuclear dimension, $$\mathcal{Z }$$ Z -stability, and algebraic simplicity for stably projectionless $$C^*$$ C ∗ -algebras

Abstract

Abstract. The main result here is that a simple separable C * -algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [33,42] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C * -algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept. IntroductionThe program to classify unital, simple, separable, nuclear C * -algebras has recently seen a small paradigm shift [13]. In light of the fact that there exist such C * -algebras that cannot be classified by ordered K-theory and traces [34], the current trend is to try to identify, using regularity properties, the C * -algebras which are (or should be) classifiable. This idea is crystallized in a conjecture (cf. [36, Section 3.5]), due to Andrew Toms and Wilhelm Winter, that among these C * -algebras, the following properties are equivalent:(i) finite nuclear dimension (defined in [44]); (ii) Z-stability (being isomorphic to one's tensor product with the Jiang-Su algebra Z, introduced in [18]); and (iii) almost unperforated Cuntz semigroup. One might also add that, modulo the UCT, conditions (i)-(iii) should be equivalent to being classifiable (though in this form, such a statement is not well-formed because classifiability is a property of a class of C * -algebras, and not a property of a single C * -algebra). At the same time, certain examples of simple, stably projectionless C * -algebras have emerged -including certain crossed products of O 2 by R [21], a nuclear, separable, non-Z-stable example [31, Theorem 4.1], and others [12,17,37]. Certain tools, old and new, already allow one to understand some of the structure of these algebras under special hypotheses [6,27]. However, most of the theory on regularity properties for C * -algebras was developed only in the unital case, and leads one to ask what obstructions (if any) exist with nonunital algebras. Certainly, the unital case carries with it a number of simplifications: for instance, the simplex of traces on a C * -algebra which take the value 1 at the unit (i.e. which are states) plays an indispensible role in the theory, and it is far from obvious what the correct replacement is in the nonunital case. Nonetheless, one hopes that the simplifications in the unital case are superficial, and that ultimately, one can find and prove analogues or generalizations of the known unital results.2010 Mathematics Subject Classification. 46L35, 46L80, 46L05, 47L40, 46L85.

Rokhlin Dimension for Flows

“…One can employ the comparison and Z-stability results of [Rob16] and [Tik14] to conclude that the crossed product itself (and not just a matrix algebra over its unitization) contains a nontrivial projection.…”

Equivariant ⁎-homomorphisms, Rokhlin constraints and equivariant UHF-absorption

“…It follows from [28, Remark 3 (ii)], and by [28,Corollary 4 (b)], [12, Corollary 6.7], and [33,Corollary 8.6], respectively, that the functors Cu ∼ and Cu are equivalent when restricted to the class of C*-algebras that are inductive limits of 1-dimensional NCCW-complexes which are either unital or simple and with trivial K 0 -group. Hence, for these classes of C*-algebras, the theorem holds when Cu ∼ is replaced by Cu.…”

Quasidiagonality of nuclear C^*-algebras

“…Since a+ isn't the unit, x(1 − a+)x * = 0 for some x ∈ A, so that τA(x(1 − a+)x * ) > 0. Hence τA(a+) < 1 (see [70,Proposition 2.11], for example) so that τA∼ (1 − a) ≥ 1 − τA(a+) > 0. . For each n, let q I,n ∈ Q denote the spectral projection of k n corresponding to the interval I, and then set q I to be the element of Q ω represented by (q I,n ).…”