We study semiprojective, subhomogeneous C*-algebras and give a detailed description of their structure. In particular, we find two characterizations of semiprojectivity for subhomogeneous C*-algebras: one in terms of their primitive ideal spaces and one by means of special direct limit structures over one-dimensional NCCW complexes. These results are obtained by working out several new permanence results for semiprojectivity, including a complete description of its behavior with respect to extensions by homogeneous C*-algebras.
We give a complete description of which unital graph C*-algebras are semiprojective, and use it to disprove two conjectures by Blackadar. To do so, we perform a detailed analysis of which projections are properly infinite in such C*-algebras.
It is proved that classifiable simple separable nuclear purely infinite C*-algebras having finitely generated K-theory and torsion-free K_1 are semiprojective. This is accomplished by exhibiting these algebras as C*-algebras of infinite directed graphs.
We prove that a simple, separable, nuclear, purely infinite classifiable $C^*$-algebra is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.
We show that a Kirchberg algebra is semiprojective if and only if it is KK-semiprojective. In particular, this shows that a Kirchberg algebra in the UCT-class is semiprojective if and only if its K-theory is finitely generated, thereby giving a positive solution to a longstanding conjecture by Blackadar.
We define equivariant semiprojectivity for C*-algebras equipped with actions of compact groups. We prove that the following examples are equivariantly semiprojective: arbitrary finite dimensional C*-algebras with arbitrary actions of compact groups; the Cuntz algebras ${\mathcal{O}}_d$ and extended Cuntz algebras $E_d,$ for finite $d,$ with quasifree actions of compact groups; the Cuntz algebra ${\mathcal{O}}_{\infty}$ with any quasifree action of a finite group. For actions of finite groups, we prove that equivariant semiprojectivity is equivalent to a form of equivariant stability of generat…
The equivariant version of semiprojectivity was recently introduced by the first author. We study properties of this notion, in particular its relation to ordinary semiprojectivity of the crossed product and of the algebra itself. We show that equivariant semiprojectivity is preserved when the action is restricted to a cocompact subgroup. Thus, if a second countable compact group acts semiprojectively on a C*-algebra $A$, then $A$ must be semiprojective. This fails for noncompact groups: we construct a semiprojective action of the integers on a nonsemiprojective C*-algebra. We also study equiv…
Let $n$ be a natural number. Recall that a C*-algebra is said to be $n$-subhomogeneous if all its irreducible representations have dimension at most $n$. In this short note, we give various approximation properties characterising $n$-subhomogeneous C*-algebras.
We show that finitely generated subhomogeneous C*-algebras have finite decomposition rank. As a consequence, any separable ASH C*-algebra can be written as an inductive limit of subhomogeneous C*-algebras each of which has finite decomposition rank. It then follows from work of H. Lin and of the second named author that the class of simple unital ASH algebras which have real rank zero and absorb the Jiang-Su algebra tensorially satisfies the Elliott conjecture.
It is shown that every Jiang-Su stable approximately subhomogeneous C*-algebra has finite decomposition rank. Previously, it was not even known that such algebras have finite nuclear dimension. A key step in the proof is that subhomogeneous C*-algebra are locally approximated by a certain class of more tractable subhomogeneous algebras, namely, a non-commutative generalization of the class of cell complexes. The result is applied to show that Jiang-Su stable minimal Z-crossed products have finite decomposition rank.
We initiate the study of pointed approximative absolute neighborhood retracts. Our motivation is to generate examples of C*-algebras that behave in unexpected ways with respect to weak semiprojectivity. We consider both weak semiprojectivity (WSP) and weak semiprojectivity with respect to the class of unital C*-algebras (WSP1). For a non-unital C*-algebra, these are different properties. One example shows a C*-algebra can fail to be WSP while its unitization is WSP. Another example shows WSP1 is not closed under direct sums.
A new category of topological spaces with additional structures, called m-towers, is introduced. It is shown that there is a covariant functor which establishes a one-to-one correspondences between unital (resp. arbitrary) subhomogeneous C*-algebras and proper (resp. proper pointed) m-towers of finite height, and between all *-homomorphisms between two such algebras and morphisms between m-towers corresponding to these algebras.
Suppose A is a unital subhomogeneous C*-algebra. We show that every central sequence in A is hypercentral if and only if every pointwise limit of a sequence of irreducible representations is multiplicity free. We also show that every central sequence in A is trivial if and only if every pointwise limit of irreducible representations is irreducible. We also give a nice repesentation of the latter algebras.
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 C (X, M_n) 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 A is a separable C*-algebra whose irreducible representations all h…
Given a compact, metric space X, we show that the commutative C*-algebra C(X) is semiprojective if and only if X is an absolute neighborhood retract of dimension at most one. This confirms a conjecture of Blackadar. Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C*-algebras. As further application of our findings we verify two conjectures of Loring and Blackadar in the commutative case, and we give a partial answer to the question, when a commutative C*-algebra is weakly (semi-)projective.
In this paper, we introduce stable recursive subhomogeneous algebras (SRSHAs), which is analogous to recursive subhomogeneous algebras (RSHAs) introduced by N. C. Phillips in the studies of free minimal integer actions on compact metric spaces. The difference between the stable version and the none stable version is that the irreducible representations of SRSHAs are infinite dimensional, but the irreducible representations of the RSHAs are finite dimensional. While RSHAs play an important role in the study of free minimal integer actions on compact metric spaces, SRSHAs play an analogous role …
A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely the class of simple C*-algebras which are inductive limits of continuous-trace C*-algebras whose building blocks have their spectrum homeomorphic to the interval [0,1] or to a finite disjoint union of closed intervals. In particular, a classification of those stably AI algebras which are inductive limits of hereditary sub-C*-algebras of interval algebras is obtained. Also, the range of the invariant is calculated.
Let $p$ be a polynomial in one variable whose roots either all have multiplicity more than 1 or all have multiplicity exactly 1. It is shown that the universal $C^*$-algebra of a relation $p(x)=0$, $\|x\| \le 1$ is semiprojective. In the case of all roots multiple it is shown that the universal $C^*$-algebra is also residually finite-dimensional. Applications to polynomially compact operators are given.
We compute the generator rank of a subhomgeneous C*-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every Z-stable ASH-algebra has generator rank one, which means that a generic element in such an algebra is a generator. This leads to a strong solution of the generator problem for classifiable, simple, nuclear C*-algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear C*-algebras.
We introduce a class of recursive subhomogeneous algebras that we call diagonal subhomogeneous and we give a notion of diagonal maps between these algebras. We show that any simple limit of diagonal subhomogeneous algebras with diagonal maps has stable rank one. As an application we show that for any minimal homeomorphism of a compact Hausdorff space the associated crossed product has stable rank one.
We analyze the decomposition rank (a notion of covering dimension for nuclear $C^*$-algebras introduced by E. Kirchberg and the author) of subhomogeneous $C^*$-algebras. In particular we show that a subhomogeneous $C^*$-algebra has decomposition rank $n$ if and only if it is recursive subhomogeneous of topological dimension $n$ and that $n$ is determined by the primitive ideal space. As an application, we use recent results of Q. Lin and N. C. Phillips to show the following: Let $A$ be the crossed product $C^*$-algebra coming from a compact smooth manifold and a minimal diffeomorphism. Then th…
Let A be a unital simple direct limit of recursive subhomogeneous C*-algebras with no dimension growth. We give criteria which specify exactly when A has real rank zero, and exactly when A has the Property (SP): every nonzero hereditary subalgebra of A contains a nonzero projection. Specifically, A has real rank zero if and only if the natural map from K_0 (A) to the continuous affine functions on the tracial state space has dense range, A has the Property (SP) if and only if the range of this map contains strictly positive functions with arbitrarily small norm. By comparison with results for …
This paper explores the following regularity properties and their relationships for simple, not-necessarily-unital C*-algebras: (i) Jiang-Su stability, (ii) Unperforation in the Cuntz semigroup, and (iii) slow dimension growth (applying only in the case that the C*-algebra is approximately subhomogeneous). An example is given of a simple, separable, nuclear, stably projectionless C*-algebra whose Cuntz semigroup is not almost unperforated. This example is in fact approximately subhomogeneous. It is also shown that, in contrast to this example, when an approximately subhomogeneous simple C*-alg…
We consider unital simple inductive limits of generalized dimension drop C*-algebras They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop $C^*$-algebras. Suppose that $A$ is one of these C*-algebras. We show that $A\otimes Q$ has tracial rank no more than one, where $Q$ is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let $A$ and $B$ be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then $A\cong B$ if and only if they have the same Elliott invariant.
We show that semiprojectivity of a C*-algebra is preserved when passing to C*-subalgebras of finite codimension. In particular, any pullback of two semiprojective C*-algebras over a finite-dimensional C*-algebra is again semiprojective.
We prove the following results for a unital simple direct limit $A$ of recursive subhomogeneous algebras with no dimension growth: (1) A has stable rank 1. (2) The projections in $M_{\infty} (A)$ satisfy cancellation: if $e \oplus q \sim f \oplus q$, then $e \sim f$. (3) $A$ satisfies Blackadar's Second Fundamental Comparability Question: if $p, q \in M_{\infty} (A)$ are projections such that $τ(p) < τ(q)$ for all normalized traces $τ$ on $A$, then $p$ is equivalent to a subprojection of $q$. (4) $K_0 (A)$ is unperforated for the strict order: if $η\in K_0 (A)$ and there is $n > 0$ such that $…
Let $(X, Γ)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $Γ$ is a discrete amenable group. It is shown that, if $(X, Γ)$ has a version of Rokhlin property (uniform Rokhlin property) and if $\mathrm{C}(X)\rtimesΓ$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes Γ$ is at most half of the mean topological dimension of $(X, Γ)$. These two conditions are shown to be satisfied if $Γ= \mathbb Z$ or if $(X, Γ)$ is an extension of a free Cantor system and $Γ$ has subexponential growth. …
Let $A$ be a separable, unital, simple C*-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of $A$ is realized as the rank of an operator in the stabilization of $A$. Assuming moreover that $A$ has locally finite nuclear dimension, we deduce that $A$ is $\mathcal{Z}$-stable if and only if it has strict comparison of positive elements. In particular, the Toms-Winter conjecture holds for separable, unital, simple, approximately subhomogeneous C*-algebras with stable rank one.
We introduce a subclass of recursive subhomogeneous algebras, in which each of the pullback maps is diagonal in a suitable sense. We define the notion of a diagonal map between two such algebras and show that every simple inductive limit of these algebras with diagonal bonding maps has stable rank one. As an application, we prove that for any infinite compact metric space $T$ and minimal homeomorphism $h\colon T\to T$, the associated dynamical crossed product $\mathrm{C^*}(\mathbb{Z},T,h)$ has stable rank one. This affirms a conjecture of Archey, Niu, and Phillips. We also show that the Toms-W…
We examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are s…
