The stable and the real rank of Z-absorbing C*-algebras

By M. Rørdam. 2004

304 citations recorded by the source corpus.

Suppose that A is a C*-algebra for which A is isomorphic to A tensor Z, where Z is the Jiang-Su algebra: a unital, simple, stably finite, separable, nuclear, infinite dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is weakly unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then A is isomorphic to A tensor O_infty if and only if A is traceless. (iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite. We also characterise when A is of real rank zero.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3afc82efe9c9ef1e008411dcd416ee5d3dc5be87

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Citations in the corpus

Central Sequences in C*-Algebras and Strongly Purely Infinite Algebras (2006, background): “Rørdam [14] it follows that D⊗Z has stable rank one and D⊗Z has real rank zero if K0(D) 6∼= Z.”
Nuclear dimension of simple stably projectionless C∗-algebras (2019, background): “The key ingredient in the proof is that W (A) is almost unperforated whenever A is Z-stable, which is due to Rørdam [54].”
Strict comparison and $$ \mathcal{Z} $$-absorption of nuclear C∗-algebras (2011, unknown): “This theorem is one of technical cornerstones in the proof of Kirchberg’s celebrated embedding theorem for exact C∗-algebras [7, 8].”
Purely infinite C*-algebras of real rank zero (2006, background): “This has been verified in [20] in the case where A in addition is exact and tensorially absorbs the Jiang-Su algebra Z.”
Rokhlin actions and self-absorbing C*-algebras (2005, background): “A ∼= A ⊗ D), and such algebras seem to enjoy nice regularity properties — see [Rørdam 1991; 1992] for absorption of UHF algebras, and [Gong, Jiang and Su 2000; Rørdam 2004; Toms and Winter 2005] for absorption of the Jiang–Su algebra.”
Dimension, comparison, and almost finiteness (2017, background): “The implications (i)⇒(ii) and (ii)⇒(iii) are theorems of Winter [53] and Rørdam [36], respectively.”
Nuclear dimension and $$\mathcal Z$$Z-stability (2014, background): “The Jiang-Su algebra has strict comparison, and stable rank one ([29], which shows the same for all simple, separable, unital, exact and stably finite Z-stable C-algebras).”
A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group (2014, background): “Under the assumptions of the lemma, it is known that K0(D) is torsion free (by [17]) and that D has cancellation of full projections by [19] and [15].”
C*-algebras of stable rank one and their Cuntz semigroups (2018, unknown): “The equivalence between (1) and (3) is proven in [ERS11, Proposition 6.2] (see also[Rør04, Corollary 4.7]).”