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 have dimension at most N (for some finite N), and if for each n the space of n-dimensional irreducible representations has finite covering dimension, then A is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it.
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 …
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 …
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 $n$ 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 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.
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 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.
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.
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 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…
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…
A unital $C^*$-algebra is called $N$-subhomogeneous if its irreducible representations are finite dimensional with dimension at most $N$. We extend this notion to operator systems, replacing irreducible representations by boundary representations. This is done by considering the matrix state space associated with an operator system and identifying the boundary representations as absolute matrix extreme points. We show that two $N$-subhomogeneous operator systems are completely order equivalent if and only if they are $N$-order equivalent. Moreover, we show that a unital $N$-positive map into a…
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.
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. …
We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra $A$ generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, $A$ satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its Elliott invariant and is approximately subhomogeneous. We give a large class of irreducible representations of nilpotent groups (of arbitrarily large nilpotency class) that satisfy the universal coefficie…
Recently, M. Daws introduced a notion of co-representation of abelian Hopf--von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf--von Neumann algebras. The key is our observation that, for a von Neumann algebra $\M$ and a reflexive operator space $E$, the normal spatial tensor product $\M \bar{\tensor} \CB(E)$ is a Banach algebra if and only if $\M$ is subhomogeneous or $E$ is completely isomorphic to column Hilbert space.
It is shown that a C*-algebra of the form C(X,U), where U is a UHF algebra, is not an inductive limit of subhomogeneous C*-algebras of topological dimension less than that of X. This is in sharp contrast to dimension-reduction phenomenon in (i) simple inductive limits of such algebras, where classification implies low-dimensional approximations, and (ii) when dimension is measured using decomposition rank, as the author and Winter proved that dr(C(X,U)) $\leq$ 2.
We introduce a notion of covering dimension for Cuntz semigroups of C*-algebras. This dimension is always bounded by the nuclear dimension of the C*-algebra, and for subhomogeneous C*-algebras both dimensions agree. Cuntz semigroups of Z-stable C*-algebras have dimension at most one. Further, the Cuntz semigroup of a simple, Z-stable C*-algebra is zero-dimensional if and only if the C*-algebra has real rank zero or is stably projectionless.
We introduce the growth rank of a C*-algebra, a (N \cup {\infty})-valued invariant which measures how far an algebra is from absorbing the Jiang-Su algebra Z tensorially. We prove that its range is exhausted by simple nuclear C*-algebras, and obtain in the process a well developed theory of unbounded dimension growth for approximately homogeneous (AH) algebras. Another consequence of the range result is the existence of a simple, nuclear, and non-Z-stable C*-algebra which is not tensorially prime. The properties of the growth rank suggest a universal property which may be considered inside any…
We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this param…
This paper studies the problems of embedding and isomorphism for countably generated Hilbert C*-modules over commutative C*-algebras. When the fibre dimensions differ sufficiently, relative to the dimension of the spectrum, we show that there is an embedding between the modules. This result continues to hold over recursive subhomogeneous C*-algebras. For certain modules, including all modules over $C_0(X)$ when $dim X \leq 3$, isomorphism and embedding are determined by the restrictions to the sets where the fibre dimensions are constant. These considerations yield results for the Cuntz semigr…
The article is devoted to investigation of classes of functions monotone as functions on general $C^*$-algebras that are not necessarily the $C^*$-algebras of all bounded linear operators on a Hilbert space as it is in classical case of matrix and operator monotone functions. We show that for general $C^*$-algebras the classes of monotone functions coincide with the standard classes of matrix and operator monotone functions. For every class we give exact characterization of $C^*$-algebras that have this class of monotone functions, providing at the same time a monotonicity characterization of …
We show that for a large class of C*-algebras $\mathcal{A}$, containing arbitrary direct limits of separable type I C*-algebras, the following statement holds: If $A\in \mathcal{A}$ and $B$ is a simple projectionless C*-algebra with trivial K-groups that can be written as a direct limit of a system of (nonunital) recursive subhomogeneous algebras with no dimension growth then the stable rank of $A\otimes B$ is one. As a consequence we show that if $A\in \mathcal A$ then the stable rank of $A\otimes\mathcal W$ is one. We also prove the following stronger result: If $A$ is separable C*-algebra t…
We calculate the real rank and stable rank of CCR algebras which either have only finite dimensional irreducible representations or have finite topological dimension. We show that either rank of A is determined in a good way by the ranks of an ideal I and the quotient A/I in four cases: When A is CCR; when I has only finite dimensional irreducible representations; when I is separable, of generalized continuous trace and finite topological dimension, and all irreducible representations of I are infinite dimensional; or when I is separable, stable, has an approximate identity consisting of proje…
In this article I study a number of topological and algebraic dimension type properties of simple C*-algebras and their interplay. In particular, a simple C*-algebra is defined to be (tracially) (m,\bar{m})-pure, if it has (strong tracial) m-comparison and is (tracially) \bar{m}-almost divisible. These notions are related to each other, and to nuclear dimension. The main result says that if a separable, simple, nonelementary, unital C*-algebra A with locally finite nuclear dimension is (m,\bar{m})-pure, then it absorbs the Jiang-Su algebra Z tensorially. It follows that A is Z-stable if and on…
A $C^{*}$-algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper, we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.
Bosons and fermions are often written by elements of other algebras. M. Abe gave a recursive realization of the boson by formal infinite sums of the canonical generators of the Cuntz algebra ${\cal O}_{\infty}$. We show that such formal infinite sum always makes sense on a certain dense subspace of any permutative representation of ${\cal O}_{\infty}$. In this meaning, we can regard as if the algebra ${\cal B}$ of bosons was a unital $*$-subalgebra of ${\cal O}_{\infty}$ on a given permutative representation by keeping their unboundedness. By this relation, we compute branching laws arising fr…
Let A be an approximately subhomogeneous (ASH) C*-algebra with slow dimension growth. We prove that if A is unital and simple, then the Cuntz semigroup of A agrees with that of its tensor product with the Jiang-Su algebra Z. In tandem with a result of W. Winter, this yields the equivalence of Z-stability and slow dimension growth for unital simple ASH algebras. This equivalence has several consequences, including the following classification theorem: unital ASH algebras which are simple, have slow dimension growth, and in which projections separate traces are determined up to isomorphism by th…
The main result here is that a simple separable C*-algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [Toms, "K-theoretic rigidity and slow dimension growth"; Winter, "Nuclear dimension and Z-stability of pure C*-algebras"] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C*-algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of…
We show that, if A is a separable simple unital C*-algebra which absorbs the Jiang-Su algebra Z tensorially and which has real rank zero and finite decomposition rank, then A is tracially AF in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of C*-algebras as above which, additionally, satisfy the Universal Coefficients Theorem. In particular, such algebras are completely determined by their ordered K-theory. They are approximately homogeneous of topological dimension less than or equal to 3, approximately sub…
We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar's Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimensio…
Embeddings of the CAR (canonical anticommutation relations) algebra of fermions into the Cuntz algebra ${\cal O}_2$ (or ${\cal O}_{2d}$ more generally) are presented by using recursive constructions. As a typical example, an embedding of CAR onto the U(1)-invariant subalgebra of ${\cal O}_2$ is constructed explicitly. Generalizing this construction to the case of ${\cal O}_{2^p}$, an embedding of CAR onto the U(1)-invariant subalgebra of ${\cal O}_{2^p}$ is obtained. Restricting a permutation representation of the Cuntz algebra, we obtain the Fock representation of CAR. We apply the results to…
Based on an embedding formula of the CAR algebra into the Cuntz algebra ${\mathcal O}_{2^p}$, properties of the CAR algebra are studied in detail by restricting those of the Cuntz algebra. Various $\ast$-endomorphisms of the Cuntz algebra are explicitly constructed, and transcribed into those of the CAR algebra. In particular, a set of $\ast$-endomorphisms of the CAR algebra into its even subalgebra are constructed. According to branching formulae, which are obtained by composing representations and $\ast$-endomorphisms, it is shown that a KMS state of the CAR algebra is obtained through the a…
Based on our previous work on the recursive fermion system in the Cuntz algebra, it is shown that a nonlinear transformation group of the CAR fermion algebra is induced from a $U(2^p)$ action on the Cuntz algebra ${\cal O}_{2^p}$ with an arbitrary positive integer $p$. In general, these nonlinear transformations are expressed in terms of finite polynomials in generators. Some Bogoliubov transformations are involved as special cases.
