Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra
By Eberhard Kirchberg, M. Rørdam. 2012
143 citations recorded by the source corpus.
We study properties of the central sequence algebra of a C*-algebra, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C*-algebra, whose trace simplex is finite dimensional, tensorially absorbs the Jiang-Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C*-algebra of having the strict comparison property to a weaker property, that we call local weak comparison. Namely, we prove that a unital separable simple nuclear C*-algebra, whose trace simplex has finite dimensional closed extreme boundary, tensorially absorbs the Jiang-Su algebra if and only if it has the local weak comparison property. We can also eliminate the nuclearity assumption, if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II_1-factor representation of the C*-algebra is a McDuff factor.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e306ddbcfae474bcee0a0adcfbb3b9b5ebe858ac
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Dimension, comparison, and almost finiteness (2017, background): “⇒(ii) when the set of extremal tracial states is finite and nonempty [30], and this was later generalized to the case where the extreme traces form a nonempty compact set with finite covering dimension [24, 42, 50]. The implication (ii)⇒(i) was first established by Sato, White, and Winter in the case o”
- Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, methodology): “, it is nonempty and its extreme boundary ∂eT (A) is compact) and ∂eT (A) has finite covering dimension ([48, 75, 85]) — these extend the work [56] which handles the case of finitely many extremal traces.”
- Dixmier approximation and symmetric amenability for C*-algebras (2013, methodology): “Hence it converges to a∈Msuch that πλ(a) = f(λ) for every λ∈K.14 NARUTAKAOZAWA The following is a W∗-analogue of the result for C∗-algebras in [HRW], and is essentially the same as Proposition 7.7 in [KR]. Corollary 12. Let M be a continuous W∗-bundle over K. Assume that each fiber πλ(M) has the McDu”
- Almost Finiteness and the Small Boundary Property (2018, unknown): “6 in [28].”
- Decomposition rank of UHF-absorbing c* -algebras (2013, methodology): “ists a positive contraction a∈ Asuch that ad∗ i xdja≈ε/N2 ϕ(d∗ixdj)a2 for x∈ F and ϕ(a) = 1. Because T(B) has compact extreme boundary with finite topological dimension, by [29, Theorem 1.1] (see also [12, 31]), we obtain a unital embedding of MN into M∩ B′. By Theorem 5 3.1 we obtain a sequence of c”
- Ranks of Operators in Simple $$C^*$$-algebras with Stable Rank One (2017, result): “)’ holds in general; see Remark 9.11. The implication ‘(3)⇒(2)’ was shown to hold under the additional assumption that ∂eT(A) is compact and finite dimensional by independent works of KirchbergRørdam, [KR14], Sato, [Sat12], and Toms-White-Winter, [TWW15], extending the work of Matuia-Sato, [MS12], th”
- Nuclear dimension of simple $$\mathrm {C}^*$$-algebras (2019, unknown): “omomorphism φ: A→ Bis (k · k2,T(A),k · k2,T(B))- contractive (since for τ∈ T(B), τ◦φis a contractive tracial functional). The (semi)norm k · k2,T(A) plays a notable role in recent structural results ([43, 40, 47, 3]), where it is often denoted k·k2,u. 1.3. Ultrapowers. Throughout, ωwill stand for a ”