Strongly self-absorbing C*-algebras

By Andrew S. Toms, W. Winter. 2005

295 citations recorded by the source corpus.

Say that a separable, unital C*-algebra V ? C is strongly self absorbing if there exists an isomorphism V such that lx> ai>e approximately unitarily equivalent *-homomorphisms. We study this class of algebras, which includes the Cuntz algebras ?2, Ooo, the UHF algebras of infinite type, the Jiang-Su algebra Z and tensor products of ?00 with UHF algebras of infinite type. Given a strongly self-absorbing C*-algebra V we characterise when a separable C*-algebra absorbs V tensorially (i.e., is P-stable), and prove closure properties for the class of separable Testable C* algebras. Finally, we compute the possible If-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C*-algebras.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4327c948e2c1c35ccff8f39f747ea8b4a6ff5115

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Citations in the corpus

Localizing the Elliott conjecture at strongly self-absorbing C*-algebras (2007, background): “…between tensor products with generalized dimension drop intervals and show how such an isomorphism may be used to obtain an asymptotic unitary intertwining between tensor products with Z. 4.1 Notation: For 0 m ≤ n ∈ N and unital C∗-algebrasA1, . . . , Am, B1, . . . , Bn let i1, . . .…”
Decomposition rank and Z-stability (2008, background): “, Z = lim → Zpν,qν , where Zpν,q ν= {f∈ C([0,1],Mp ν,qν ) | f(0) ∈ Mpν ⊗1Mq and f(1) ∈ 1Mq ⊗Mqν },ν∈ N, and pν and qν are prime. It was shown in [16] that Z is strongly self-absorbing in the sense of [33], and that it is KK-equivalent to the complex numbers. In [30], several alternative characteriza”
The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension (2016, background): “4 of [50], in order to show that A is Q-stable, it is enough to show that A⊗Zp,q is Q-stable.”
Quasidiagonality of nuclear C*-algebras (2015, background): “n a fashion which picks up the canonical trace on C∗ r(G).33 The result then follows from Corollary 6.5. Strongly self-absorbing C∗-algebras, whose study from an abstract perspective was initiated in [72], are omnipresent in the structure and classiﬁcation theory of nuclear C∗-algebras. These are si”
On the Classification of Simple Z‐Stable C*‐Algebras with Real Rank Zero and Finite Decomposition Rank (2005, background): “2 The Jiang–Su algebra is strongly self-absorbing in the sense of [19].”
Nuclear dimension and -stability of pure C ∗ -algebras (2010, background): “By [31] there is an isomorphism φ : A → A ⊗ Z”
Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra (2012, result): “We have learned that Andrew Toms, Stuart White, and Wilhelm Winter independently have obtained results very similar to ours, see [25].”
Rokhlin actions and self-absorbing C*-algebras (2005, methodology): “In [Toms and Winter 2007], it was shown that if D is strongly self-absorbing and K1-injective (i.e., the MSC2000: 46L05, 46L55.”
On the classification problem for nuclear C^*-algebras (2005, background): “There is evidence that slow dimension growth and Z-stability are equivalent for ASH algebras — in the case of simple and unital AH algebras with unique trace this has recently been proved ([ TW1 ], [TW2]).”
Regularity properties in the classification program for separable amenable C*-algebras (2007, background): “It is very robust with respect to common constructions (see [59]).”
Strongly self-absorbing *-dynamical systems (2015, methodology): “…b = 1 B in Lemma 2.3, we obtain the following unital variant: Let B be unital and let We are now going to prove an equivariant McDuﬀ-type absorption result in the unital case, modelling the well-known McDuﬀ-type results [47, 7.2.2], [50, 2.3] or [26, 4.11] from the non-equivariant setting.”
Strongly self-absorbing C*-algebras are Z-stable (2009, background): “Introduction A separable unital C∗-algebra D 6= C is called strongly self-absorbing, if there is an isomorphism D → D ⊗ D which is approximately unitarily equivalent to the ﬁrst factor embedding, cf. [12]. The interest in such algebras largely arises from Elliott’s program to classify nuclear C∗-alg”