The Jiang–Su algebra revisited

By M. Rørdam, W. Winter. 2008

173 citations recorded by the source corpus.

Abstract We give a number of new characterizations of the Jiang–Su algebra 𝒵, both intrinsic and extrinsic, in terms of C*-algebraic, dynamical, topological and K-theoretic conditions. Along the way we study divisibility properties of C*-algebras, we give a precise characterization of those unital C*-algebras of stable rank one that admit a unital embedding of the dimension-drop C*-algebra Z n, n+1, and we prove a cancellation theorem for the Cuntz semigroup of C*-algebras of stable rank one.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/89d5bebae8308a759afd7d5c9957b4394d8f738f

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In Algebra and Number Theory

1. W. Winter
2. Andrew S. Toms
3. Z. Niu
4. G. Elliott
5. Stuart White
6. Leonel Robert
7. Huaxin Lin
8. Samuel Evington

In Mathematical Physics

1. W. Winter
2. Andrew S. Toms
3. Z. Niu
4. G'abor Szab'o
5. Yasuhiko Sato
6. Leonel Robert
7. Stuart White
8. G. Elliott

Citations in the corpus

• K-Theory for operator algebras. Classification of C$^*$-algebras (2009, background): “As remarked in [58], it is not true that the Cuntz semigroup is cancellative, even in the stable rank one case.”
• Nuclear dimension of simple $$\mathrm {C}^*$$-algebras (2019, background): “ability of a II1 factor in terms of approximately central matrix algebras ([45]), Z-stability is equivalent to the existence of approximately central matrix cones which are large in a suitable sense ([53]). Moreover, for simple, nuclear C∗-algebras, via the groundbreaking work of Matui and Sato ([43”
• Tracially Z-absorbing C*-algebras (2012, unknown): “order zero maps φ :Mn → A and ψ :M2 :→ A such that the image of any normalized matrix under φ or ψ commutes up to ε with the elements of F , and such that ψ(e1,1)φ(e1,1) = ψ(e1,1) and ψ(e2,2) = 1A − φ(1) (see [20]).”
• C*-algebras of stable rank one and their Cuntz semigroups (2018, unknown): “This is proven in [RW10, Theorem 4.3] for the ‘non-complete’ version of the Cuntz semigroup W ( A ).”
• Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture (2009, methodology): “We refer the reader to [4] and to [20] for a complete discussion of the Jiang-Su algebra and its relevance to Elliott’s program.”
• Nuclear dimension and -stability of pure C ∗ -algebras (2010, background): “In [29], several alternative characterizations of the Jiang–Su algebra were given; see also [10, 41].”
• The Rohlin property for automorphisms of the Jiang-Su algebra (2009, methodology): “ne the projectionless C∗- algebra Zk for k ∈ N\{1} by Zk = {f ∈ C([0,1]) ⊗Mk∞ ⊗M(k+1)∞; f(0) ∈ Mk∞ ⊗1k+1∞, f(1) ∈ 1k∞ ⊗Mk+1∞}. This projectionless C∗-algebra Zk was introduced by Rørdam and Winter in [30], [34] as a mediator between C∗-algebras absorbing UHF algebras and C∗- algebras absorbing the J”
• Tensor products and regularity properties of Cuntz semigroups (2014, unknown): “46057 [86] Rørdam, Mikael; Winter, Wilhelm, The Jiang-Su algebra revisited, J.”
• Decomposition rank and Z-stability (2008, unknown): “ F= F(0) ⊕...⊕F(m). We have kϕ¯ψ¯(f) −fk ≤ kϕ¯ψ¯(f)−ϕ′ψ′(f)k +kϕ′ψ′(f) −fk (31) ≤ kψ¯(f)−ψ′(f)k+kϕ′ψ′(f) −fk (32) < 2η; moreover, kϕ¯ψ¯(f2)−f2k (31) = kϕ′(pψ′(f2)p)−f2k 1. < kϕ′(pψ′(f)2p)−f2k+η (30) ≤ kϕ′(ψ′(f)2) −f2k+2η 1. < kϕ′ψ′(f2) −f2k+3η (33) < 4η, where for the last estimates of (”
• Pullbacks, $C(X)$-algebras, and their Cuntz semigroup (2011, background): “This construction can be slightly simplified using a unique dimension drop algebra Zpq and a unique morphism γ : Zpq → Zpq but allowing p,q to be supernatural numbers of infinite type, that is to say, p∞ i = pi (see [28]).”
• The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension (2016, background): “A combination of [46] [47] and [19] shows that A also satisfies the rest of conditions of A.”
• Classification of inductive limits of 1-dimensional NCCW complexes (2010, background): “If A⊗K is projectionless, a different route in the computation of Cu∼(A) must be followed, since there is currently no version of Winter’s result for projectionless C∗-algebras (although such a result is likely to be true).”