Covering Dimension of C*-Algebras and 2-Coloured Classification

Abstract

We introduce the concept of finitely coloured equivalence for unital * -homomorphisms between C * -algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *homomorphisms from separable, unital, nuclear C * -algebras into ultrapowers of simple, unital, nuclear, Z-stable C * -algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data.As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C * -algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a "homotopy equivalence implies isomorphism" result for large classes of C * -algebras with finite nuclear dimension.

Almost Finiteness and the Small Boundary Property

“…While towers 1 The first proof of this fact was given by Connes as an application of his celebrated result [7] that injectivity implies hyperfiniteness, whose full force is still needed to prove hyperfiniteness of the group von Neumann algebra itself. 2 Whether or not this subequivalence is itself implemented in an approximately central way is roughly what separates Z -stability from its specialization to the nuclear setting. 3 Nuclearity is automatic in this case since the acting group is amenable.…”

The Toeplitz algebra has nuclear dimension one

“…On the one hand, we do not know in general whether nuclear dimension is sub‐additive with respect to tensor products. On the other hand, there are several situations where dimension reduces when there is enough ‘non‐commutative space’ available, in particular . In all of these, dimension reduction is quite dramatic, and eventually relies (more or less explicitly) on the existence of simple — in most cases, strongly self‐absorbing — local tensor factors, like the Cuntz algebras and , UHF algebras, or the Jiang–Su algebra .…”

“…In this context, there now is a result as complete as can be, and the crucial hypothesis is just finiteness of nuclear dimension, with no reference to the actual value. In hindsight, the latter is not too surprising, since we now know that for separable, simple, unital C * -algebras, the only possible values for the nuclear dimension are 0, 1 and infinity; see [2] and the references therein, in particular [1,8]. In the non-simple case, the nuclear dimension potentially carries more information (after all, for continuous trace C * -algebras, it coincides on the nose with covering dimension of the spectrum).…”

“…Next, recall that by [6, Remark 2.4] order zero maps out of finite-dimensional C * -algebras into quotient C * -algebras always lift to order zero maps. As a consequence, there are completely positive contractive order zero liftsφ (1)…”

“…note thatφ (1) k is a completely positive order zero contraction since each of the two summands is, and since they map into the orthogonal algebras C n k and A n k .…”