Large subalgebras

By N. Christopher Phillips. 2014

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We define and study large and stably large subalgebras of simple unital C*-algebras. The basic example is the orbit breaking subalgebra of a crossed product by Z, as follows. Let X be an infinite compact metric space, let h : X → X be a minimal homeomorphism, and let Y ⊂ X be closed. Let u ∈ C∗(Z, X, h) be the standard unitary. The Y -orbit breaking subalgebra is the subalgebra of C∗(Z, X, h) generated by C(X) and all elements fu for f ∈ C(X) such that f |Y = 0. If h (Y ) ∩ Y = ∅ for all n ∈ Z \ {0}, then the Y -orbit breaking subalgebra is large in C∗(Z, X, h). Large subalgebras obtained via generalizations of this construction have appeared in a number of places, and we unify their theory in this paper. We prove the following results for an infinite dimensional simple unital C*-algebra A and a stably large subalgebra B ⊂ A: • B is simple and infinite dimensional. • If B is stably finite then so is A, and if B is purely infinite then so is A. • The restriction maps T(A) → T(B) and QT(A) → QT(B) (on tracial states and quasitraces) are bijective. • When A is stably finite, the inclusion of B in A induces an isomorphism on the semigroups that remain after deleting from Cu(B) and Cu(A) all the classes of nonzero projections. • B and A have the same radius of comparison. The purpose of this paper is to define what we call a large subalgebra B in a simple unital C*-algebra A, and to show how properties of B can be used to deduce properties of A. The main applications so far are to the structure of crossed product C*-algebras, and are treated elsewhere; see the discussion below. They work because it is possible to choose large subalgebras of these crossed products which are of an accessible form, such as a direct limit of recursive subhomogeneous algebras. A strengthening of the condition (centrally large subalgebras) permits further results about the containing algebra; this will also be treated elsewhere [3], [4]. Large subalgebras (and centrally large subalgebras) are an abstraction of the Putnam subalgebra of the crossed product by a minimal homeomorphism. LetX be an infinite compact metric space, and let h : X → X be a minimal homeomorphism. Let u be the standard unitary in the crossed product C(Z, X, h). Fix y ∈ X . Then the Putnam subalgebra of C(Z, X, h) is generated by C(X) and all elements fu with f ∈ C(X) satisfying f(y) = 0. This algebra was introduced by Putnam in [32] whenX is the Cantor set. (Putnam actually used uf rather than fu, but this choice makes the relationship with Rokhlin towers more awkward.) In this case, on the one hand, the subalgebra is an AF algebra, while, on the other hand, it is closely Date: 23 August 2014. 2010 Mathematics Subject Classification. Primary 46L05; Secondary 46L55. This material is based upon work supported by the US National Science Foundation under Grants DMS-0701076 and DMS-1101742. 1 2 N. CHRISTOPHER PHILLIPS enough related to C(Z, X, h) to use information about it to obtain information about C(Z, X, h). This method was used in [22] and Section 4 of [29] to obtain information on the order onK0(C (Z, X, h)) for general finite dimensional X . The Putnam subalgebra played a key role in [20], in which it is proved that C(Z, X, h) has tracial rank zero whenever this property is consistent with its K-theory and dim(X) <∞, and in [39], which gives classifiability of such crossed products in some cases in which they don’t have tracial rank zero. The paper [39] also required a generalization in which one used two points y1 and y2 on distinct orbits of h, and in the definition used fu for f ∈ C(X) such that f(y1) = f(y2) = 0. A more recent application appears in [13]. Versions in which f is required to vanish on a larger subset are important in [19] and [36]. Further applications of such generalized Putnam algebras will appear in [16]. Particular examples of these subalgebras have been studied in their own right in [15]. The subalgebra Aθ,γ considered there (see the introduction) is large whenever the zero set of the function γ intersects each orbit at most once. Under similar conditions, the algebras studied in [37] are large in the corresponding three dimensional noncommutative tori. The abstraction to large subalgebras has four motivations. The first is the use, as described above, of subalgebras of C(Z, X, h) generated by C(X) and the elements fu with f required to vanish on a subset with more than one point, but which meets each orbit of h at most once. The second is the generalization to crossed products by automorphisms of C(X,D) in [8]. Let X be an infinite compact metric space, let h : X → X be a minimal homeomorphism, let D be a simple unital C*algebra satisfying suitable additional conditions, and let α ∈ Aut(C(X,D)) be an automorphism such that, in terms of C(X)⊗D, we have α(f⊗1) = (f ◦h−1)⊗1 for all f ∈ C(X). Let u ∈ C ( Z, C(X,D), α) be the standard unitary in the crossed product, and fix y ∈ Y . Then the subalgebra used is the one generated by C(X,D) and all fu with f ∈ C(X,D) satisfying f(y) = 0. A third, stronger, motivation for the abstraction is the construction of large subalgebras in more general crossed products, where the subalgebras don’t have convenient descriptions. Large subalgebras (without the name) play a key role in [27], where they are used to prove that if Z acts freely and minimally on the Cantor set X , then C(Z, X) has real rank zero, stable rank one, and order on projections determined by traces. It is shown in [31] that if X above is a finite dimensional compact metric space, then C(Z, X) contains a large subalgebra which is a simple direct limit, with no dimension growth, of recursive subhomogeneous C*-algebras. Although this paper is still unpublished, this was the first proof that, for such X , the crossed product has strict comparison of positive elements. A more abstract version is needed because there is no known easy description of the subalgebra; rather, there is just an existence proof. A fourth reason for the abstraction is the role played by large subalgebras in [12]. This paper considers C*-algebras obtained from irrational rotation algebras by “cutting” each of the standard unitary generators at one or more points in its spectrum, say by adding logarithms of them or adding some spectral projections. The new algebras are shown to be AF. One of the technical tools is that the original irrational rotation algebra is a large subalgebra the new algebra. In this case, the containing algebra is not even given as a crossed product.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

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