Regularity properties in the classification program for separable amenable C*-algebras

By G. Elliott, Andrew S. Toms. 2007

190 citations recorded by the source corpus.

We report on recent progress in the program to classify separable amenable C∗-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program’s successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.

Subfields: Mathematical Physics, Psychiatry and Mental health

View Paper: https://www.semanticscholar.org/paper/c63e5ac346d2d542c86099a48bbc6579dbcd684b

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Citations in the corpus

Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, background): “[27, 92]; the precise form stated here is given in [97, Conjecture 9.”
Decomposition rank and Z-stability (2008, background): “These three properties and their relationships were discussed extensively in [13].”
Nuclear dimension and -stability of pure C ∗ -algebras (2010, background): “In [35, 40, 43], some links between such regularity properties have been proven, and some others have been formally established as conjectures (see also [12]).”
Comparison Theory and Smooth Minimal C*-Dynamics (2008, background): “ful regularity property necessary for the confirmation of G. A. Elliott’s K-theoretic rigidity conjecture (see [ 11 ] and [27]).”
Completely positive maps of order zero (2009, background): “[4], [2]), at this point not even a bivariant version (resembling Kasparov’s KK-theory) has been developed.”
The Rohlin property for automorphisms of the Jiang-Su algebra (2009, methodology): “O A ] For projectionless C∗-algebras absorbing the Jiang-Su algebra tensorially, we study a kind of the Rohlin property for autmorphisms.”
Completeness of the isomorphism problem for separable C*-algebras (2013, background): “The Elliott invariant, which consists of the groups K0 and K1 together with the tracial simplex and the pairing map, was conjectured (see [19,24]) to completely classify all infinite-dimensional, separable, simple nuclear C*-algebras.”
A Remark on Invariants for C*-Algebras of Stable Rank One (2007, background): “In [5], Cuntz introduced an invariant for C*-algebras—very much analogous to the purely algebraic invariant K0—which has come to be known as the Cuntz semigroup, and which has recently received considerable attention (see [21], [2], [4], and [12]).”
The cone of lower semicontinuous traces on a C*-algebra (2008, background): “The reader is referred to [7], [10], and [20] for a more detailed discussion of the properties of Cu(A), as well as of the closely related semigroup W(A).”
K-Theory for operator algebras. Classification of C$^*$-algebras (2009, background): “Portions of these notes are shamelessly based on [9] and [50] (and also [ 25 ] in part, which constitutes a very nice survey of the whole story).”
Turbulence, orbit equivalence, and the classification of nuclear C*-algebras (2011, background): “The general theory of C*-algebras can be found in [2] and references for Elliott’s classification program are [23] and [4].”
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.”