Recursive subhomogeneous algebras

By S. »ALGEBRAS, N. Christopher Phillips. 2001

80 citations recorded by the source corpus.

We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies, in particular, that if is a separable C*-algebra whose irreducible representations all have dimension at most and if for each the space of -dimensional irreducible representations has finite covering dimension, then is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it. Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in separate papers.

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

View Paper: https://www.semanticscholar.org/paper/56cd22e9c63966b270955ebad4696e9b6c65f990

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Citations in the corpus

Strongly self-absorbing C*-algebras (2005, methodology): “jn and topological dimension ajn + inbjn, cf. also Proposition 3.4 of [ 29 ].”
Rokhlin Dimension and C*-Dynamics (2012, result): “[21], hence satisfies the hypotheses of [31, Theorem 6.”
Cancellation and stable rank for direct limits of recursive subhomogeneous algebras (2001, background): “tion: if η ∈ K0(A) satisﬁes τ∗(η) > 0 for all normalized traces τ on A, then there is a projection p ∈ M∞(A) such that η = [p]. 0. Introduction Recursive subhomogeneous algebras were introduced in [32]; we recall the deﬁ- nition below. They include ﬁnite direct sums of not necessarily trivial uni”
The C∗-algebra of a minimal homeomorphism of zero mean dimension (2014, background): “(See [13] for more details on recursive subhomogeneous C*-algebras.”
Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture (2009, background): “We refer the reader to [17] for the basic definitions and terminology related to these algebras.”
Comparison Theory and Smooth Minimal C*-Dynamics (2008, background): “Let us recall some of the terminology and results from [22].”
Pullbacks, $C(X)$-algebras, and their Cuntz semigroup (2011, background): “The class of Recursive Subhomogeneous Algebras (rsh-algebras) defined in [21] is the smallest class R of C∗-algebras which contains C(X,Mn) for all compact Hausdorff spaces X and n 1, and which is closed under isomorphisms and pullbacks of the type”
Crossed products by minimal homeomorphisms (2004, methodology): “For recursive subhomogeneous algebras and recursive subhomogeneous decompositions as in the second theorem, see [34], Section 1.”