The Jiang--Su algebra Z has come to prominence in the classification program for nuclear C*-algebras of late, due primarily to the fact that Elliott's classification conjecture predicts that all simple, separable, and nuclear C*-algebras with unperforated K-theory will absorb Z tensorially (i.e., will be Z-stable). There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and Z-stable C*-algebras. We prove that virtually all classes of nuclear C*-algebras for which the Elliott conjecture has been confirmed so far, consist of Z-stable C*-algebras. This result follows in large part from the following theorem, also proved herein: separable and approximately divisible C*-algebras are Z-stable.
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.
Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $A\otimes Q$ can be tracially approximated by unital Elliott-Thomsen algebras with trivial $\textrm{K}_1$-group, where $Q$ is the universal UHF algebra. In particular, it follows that $A$ is classifiable by the Elliott invariant if $A$ is Jiang-Su stable.
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…
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…
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.
We prove that the infinite tensor power of a unital separable C*-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C*-algebras without characters known to lie outside it. This category moreover contains the vast majority of our stock-in-trade separable amenable C*-algebras, and is closed under passage to separable superalgebras and quotients, and hence to unital tensor products, unital direct limits, and cr…
Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that $ϕ_1, ϕ_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(c)u_t=ϕ_2(c)\tforal c\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $ϕ_1^‡=ϕ_2^‡,$ $(ϕ_1)_T=(ϕ_2)_T$ and a rotation related map $\bar{R}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. Applying this result together with a result of W…
Let $A$ be a unital simple separable C*-algebra satisfying the UCT. Assume that $\mathrm{dr}(A)<+\infty$, $A$ is Jiang-Su stable, and $\mathrm{K}_0(A)\otimes \mathbb{Q}\cong \mathbb{Q}$. Then $A$ is an ASH algebra (indeed, $A$ is a rationally AH algebra).
We study the validity of the Blackadar-Kirchberg conjecture for extensions of separable, nuclear, quasidiagonal $C^*$-algebras that satisfy the UCT. More specifically, we show that the conjecture for the extension has an affirmative answer if the ideal lies in a class of $C^*$-algebras that is closed under local approximations and contains all separable ASH algebras, as well as certain classes of simple, unital $C^*$-algebras and crossed products of unital $C^*$-algebras with the integers.
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.
This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C_0(X,A), where A is a unital, simple, Z-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(T,A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.
Suppose A and B are unital C*-algebras and A is separable. Let Rep(A,B) denote the set of all unital *-homomorphisms from A to B with the topology of pointwise convergence. We consider the problem of when the closure of the unitary orbit of a single representation in Rep(A,B) is path-connected. An affirmative answer was given by the first author when A is singly generated and B is the algebra of all operators on a separable Hilbert space. We extend this result for all separable A. We also give an affirmative answer when A is AF or homogeneous and B is a von Neumann algebra or when A is ASH and…
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…
We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2.
We prove the title. This characterizes the Jiang-Su algebra Z as the uniquely determined initial object in the category of strongly self-absorbing C*-algebras.
Various notions of stable ranks are studied for topological algebras. Some partial answers to R.G. Swan's problem (Have two Banach or good Fréchet algebras as in the density theorem in K-theory the same stable rank ?) are obtained. For example, a Fréchet dense $\ast$-subalgebra $A$ of a $C^\ast$-algebra $B$, closed under $C^\infty$-functional calculus of self-adjoint elements, has the same Bass stable rank as $B$.
We establish a general result about extending a right invertible row over a Banach algebra to an invertible matrix. This is applied to the computation of right topological stable rank of a split exact sequence. We also introduce a quantitative measure of stable rank. These results are applied to compute the right (left) topological stable rank for all nest algebras. This value is either 2 or infinity, and rtsr(T(N)) = 2 occurs only when N is of ordinal type less than omega^2 and the dimensions of the atoms grows sufficiently quickly. We introduce general results on `partial matrix algebras' ov…
Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable C*-algebra is the stationary inductive limit of a C*-stable Fell algebra that has compact spectrum and trivial Dixmier-Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to in principle compute the K-theory of the stable C*-algebra. A specif…
We classify unital monomorphisms into certain simple Z-stable C^*-algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C^*-algebra, or any unital simple separable nuclear Z-stable C^*-algebra satisfying the UCT such that C\otimes B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z-stable C^*-algebra such that A\otimes B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C^*-algebra whose projections separate traces and whose extr…
We obtain an improved upper bound for the nuclear dimension of extensions of $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebras. In particular, we prove that the nuclear dimension of a full extension of an $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebra by a stable AF algebra is one.
We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras $L_Z(E_2)$ and $L_Z(E_{2-})$ are not stably *-isomorphic.
We study the general and connected stable ranks for C*-algebras. We estimate these ranks for pullbacks of C*-algebras, and for tensor products by commutative C*-algebras. Finally, we apply these results to determine these ranks for certain commutative C*-algebras, and non-commutative CW-complexes.
We study the matricial field (MF) property for certain reduced crossed product C*-algebras and their traces. Using classification techniques and induced K-theoretic dynamics, we show that reduced crossed products of ASH-algebras of real rank zero by free groups are MF if and only if they are stably finite. We also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties. Combining these results with recent progress in Elliott's Classification Program, it follows that if $A$ is a separable, simple, unital, nuclear, monotracial C*-al…
The class of AD algebras of real rank zero is classified by an exact sequence of K-groups with coefficients, equipped with certain order structures. Such a sequence is always split, and one may ask why, then, the middle group is relevant for classification. The answer is that the splitting map can not always be chosen to respect the order structures involved. This may be rephrased in terms of the ideals of the C*-algebras in question. We prove that when the C*-algebra has only finitely many ideals, a splitting map respecting these always exists. Hence AD algebras of real rank zero with finitel…
We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu's non-commutative disc algebras and to free semigroup algebras as well.
For an arbitrary countable directed graph E we show that the only possible values of the stable rank of the associated Cuntz-Krieger algebra C*(E) are 1, 2 or \infty. Explicit criteria for each of these three cases are given. We characterize graph algebras of type I, and graph algebras which are inductive limits of C*-algebras of type I. We also show that a gauge-invariant ideal of a graph algebra is itself isomorphic to a graph algebra.
We study the general and connected stable ranks for $C^{\ast}$-algebras. We estimate these ranks for certain $C(X)$-algebras, and use that to do the same for certain group $C^{\ast}$-algebras. Furthermore, we also give estimates for the ranks of crossed product $C^{\ast}$-algebras by finite group actions with the Rokhlin property.
The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.
Let A be a unital maximal full algebra of operator fields with base space the k-cube [0,1]^k and fibre algebras, say, {A_t}_{t \in [0,1]^k}. Then the stable rank of A is bounded above by the supremum of the stable ranks sr(C([0,1]^k) \otimes A_t) for t \in [0,1]^k. Using this estimate, we compute the stable ranks of the universal C^*-algebras of the discrete Heisenberg groups.
A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras. Moreover, it contains all unital simple separable amenable C*-algebras which satisfy the UCT and have finite rational tracial rank
We show that every nuclear $\mathcal O_\infty$-stable *-homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, $\mathcal O_\infty$-stable C*-algebras have nuclear dimension 1. We also characterise when $\mathcal O_\infty$-stable C*-algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full $\mathcal O_2$-stable *-homomorphisms have nuclear dimension 0.
We present a classification theorem for a class of unital simple separable amenable ${\cal Z}$-stable $C^*$-algebras by the Elliott invariant. This class of simple $C^*$-algebras exhausts all possible Elliott invariant for unital stably finite simple separable amenable ${\cal Z}$-stable $C^*$-algebras. Moreover, it contains all unital simple separable amenable $C^*$-alegbras which satisfy the UCT and have finite rational tracial rank.
We classify spectrum-preserving endomorphisms of stable continuous-trace C^*-algebras up to inner automorphism by a surjective multiplicative invariant taking values in finite dimensional vector bundles over the spectrum. Specializing to automorphisms, this gives a different approach to results of Phillips and Raeburn.
It is shown that projectionless C*-algebras that tensorially absorb the Jiang-Su algebra have the property that every element is a limit of products of two nilpotents. This is then used to classify the approximate unitary equivalence classes of positive elements in such C*-algebras using traces.
We show that the class of C*-algebras with stable rank greater than a given positive integer is axiomatizable in logic of metric structures. As a consequence we show that the stable rank is continuous with respect to forming ultrapowers of C*-algebras, and that stable rank is Kadison--Kastler stable.
A new homology is defined for a non-self-adjoint operator algebra and distinguished masa which is based upon cycles and boundaries associated with complexes of partial isometries in the stable algebra. Under natural hypotheses the zeroth order group coincides with the $\ K_0$\ group of the generated C*-algebra. Several identifications and applications are given, and in particular it is shown how stable homology is significant for the classification of regular subalgebras and regular limit algebras.
Let Z denote the simple limit of prime dimension drop algebras that has a unique tracial state. Let A != 0 be a unital C^*-algebra with A = A tensor Z. Then the homotopy groups of the group U(A) of unitaries in A are stable invariants, namely, π_i(U(A)) = K_{i-1}(A) for all integers i >= 0. Furthermore, A has cancellation for full projections, and satisfies the comparability question for full projections. Analogous results hold for non-unital Z-stable C^*-algebras.
We study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also determine exactly which finite, locally convex k-graphs yield unital stably finite C*-algebras. We give several examples to illustrate our results.
We show that every separable C*-algebra of real rank zero that tensorially absorbs the Jiang-Su algebra contains a dense set of generators. It follows that in every classifiable, simple, nuclear C*-algebra, a generic element is a generator.
We show that the tensor product of two unital C*-algebras, one of which is nuclear and admits a unital *-homomorphism from (the building blocks of) the Jiang-Su algebra, has Kadison's similarity property. As a consequence, we obtain that a unital C*-algebra which absorbs the Jiang-Su algebra tensorially also has this property.
Let G be a finitely generated, torsion-free, two-step nilpotent group. Let C^*(G) be the universal C^*-algebra of G. We show that acsr(C^*(G)) = acsr(C((\hat{G})_1)), where for a unital C^*-algebra A, acsr(A) is the absolute connected stable rank of A, and (\hat{G})_1 is the space of one-dimensional representations of G. For the case of stable rank, we have close results. In the process, we give a stable rank estimate for maximal full algebras of operator fields over a metric space.
Suppose that $A,B$ are nuclear, separable ${\rm C}^*$-algebras of stable rank one and real rank zero, $A$ is unital simple, $B$ is stable and $({\rm K}_0(B),{\rm K}_0^+(B))$ is weakly unperforated in the sense of Elliott \cite{Ell}. We show that any unital extension with trivial index maps of $A$ by $B$ is absorbing.
It is shown that, for a C*-algebra of stable rank one (i.e., in which the invertible elements are dense), two well-known isomorphism invariants, the Cuntz semigroup and the Thomsen semigroup, contain the same information. More precisely, these two invariants, viewed appropiately, determine each other in a natural way.
A necessary and sufficient condition for the simplicity of the C*-algebra reduced free product of finite dimensional abelian algebras is found, and it is proved that the stable rank of every such free product is 1. Related results about other reduced free products of C*-algebras are proved.
Trace scaling automorphisms of stable AF algebras with dimension group totally ordered are outer conjugate if the scaling factors are the same (not equal to one). This is an adaptation of a similar result for the AFD type II_infty factor by Connes and extends the previous result for stable UHF algebras.
We introduce the notion of continuous twisted partial actions of a locally compact group on a C*-algebra. With such, we construct an associated C*-algebraic bundle called the semidirect product bundle. Our main theorem shows that, given any C*-algebraic bundle which is second countable and whose unit fiber algebra is stable, there is a continuous twisted partial action of the base group on the unit fiber algebra, whose associated semidirect product bundle is isomorphic to the given one.
