Decomposition rank of approximately subhomogeneous C^*-algebras

Abstract

<jats:title>Abstract</jats:title><jats:p>It is shown that every Jiang–Su stable approximately subhomogeneous <jats:inline-formula id="j_forum-2020-0018_ineq_9999_w2aab3b7e5449b1b6b1aab1c15b1b1Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2020-0018_eq_1236.png" /><jats:tex-math>{{\mathrm{C}^{*}}}</jats:tex-math></jats:alternatives></jats:inline-formula>-algebra has finite decomposition rank. This settles a key direction of the Toms–Winter conjecture for simple approximately subhomogeneous <jats:inline-formula id="j_forum-2020-0018_ineq_9998_w2aab3b7e5449b1b6b1aab1c15b1b3Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2020-0018_eq_1236.png" /><jats:tex-math>{{\mathrm{C}^{*}}}</jats:tex-math></jats:alternatives></jats:inline-formula>-algebras. A key step in the proof is that subhomogeneous <jats:inline-formula id="j_forum-2020-0018_ineq_9997_w2aab3b7e5449b1b6b1aab1c15b1b5Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2020-0018_eq_1236.png" /><jats:tex-math>{{\mathrm{C}^{*}}}</jats:tex-math></jats:alternatives></jats:inline-formula>-algebras are locally approximated by a certain class of more tractable subhomogeneous algebras, namely a non-commutative generalization of the class of cell complexes. The result is applied, in combination with other recent results, to show classifiability of crossed product <jats:inline-formula id="j_forum-2020-0018_ineq_9996_w2aab3b7e5449b1b6b1aab1c15b1b7Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2020-0018_eq_1236.png" /><jats:tex-math>{{\mathrm{C}^{*}}}</jats:tex-math></jats:alternatives></jats:inline-formula>-algebras associated to minimal homeomorphisms with mean dimension zero.</jats:p>

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

“…There are other possible notions of coloured versions of equivalence; [16] will discuss the relationship between n-coloured equivalence and the stronger concept of n-intertwined * -homomorphisms. 31 Note that, despite its name, there is no reason to expect that (approximate) n-coloured equivalence is an equivalence relation. However it is immediate that (approximately) n-coloured equivalent * -homomorphisms agree on traces as necessarily the w (i) in the definition will satisfy n−1 i=0 w (i) w (i) * = n−1 i=0 w (i) * w (i) = 1 B .…”

“…An alternative -in fact almost orthogonal -approach to the implication (ii) =⇒ (i) of the Toms-Winter conjecture is through the examination of C * -algebras arising as inductive limits of concrete building blocks. This was initiated by AT and WW in [86], who showed that Z-stable approximately homogeneous (AH) C * -algebras have decomposition rank at most 2, and very recently extended in [31] (with the same estimate) to Z-stable ASH algebras. These results, and the dimension computations in the present paper complement each other nicely: here we require no inductive limit structure, and so our methods work under abstract axiomatic conditions; in particular we obtain nuclear dimension results in the absence of the Universal Coefficient Theorem (UCT) and quasidiagonality.…”

“…These results, and the dimension computations in the present paper complement each other nicely: here we require no inductive limit structure, and so our methods work under abstract axiomatic conditions; in particular we obtain nuclear dimension results in the absence of the Universal Coefficient Theorem (UCT) and quasidiagonality. In contrast, the results of [86,31] do not require simplicity or tracial state space restrictions, but only work for algebras with very concrete building blocks.…”

Inverse systems of groupoids, with applications to groupoid C⁎-algebras

“…preforming operations on a tensor factor of C 0 (X) for some locally compact Hausdorff space X) has had a tremendous impact to the theory of C * -algebras, including in classification. In [11], Elliot et al, as a stepping stone to proving that the decomposition rank of Z-stable subhomogeneous C * -algebras is at most 2, prove that one can locally approximate any unital subhomogenous C * -algebra by noncommutative CWcomplexes which, in the commutative case, have exactly the same topological dimension. Jiang and Su in [19], Razak in [25], Jacelon in [18], and Evans and Kishimoto in [14] successfully use interval algebras and the folding thereof to classify large classes of algebras and to create new examples of C * -algebras.…”

Extensions of quasidiagonal $C^*$-algebras and controlling the $K_0$-map of embeddings

“…5.4] to "lift" these maps to the C * -algebra level, and thus achieve the existence of the embeddings needed to show the presence of K 0 -embedding property. For the case when the ideal is a separable ASH algebra, we use results and techniques from [9], [10] and [16].…”