Z-stable Ash Algebras
نویسنده
چکیده
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.
منابع مشابه
Algebras with Locally Finite Decomposition
We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C∗-algebras. The concept is particularly relevant for Elliott’s program to classify nuclear C∗algebras by K-theory data. We study some of its properties and show that a simple unital C∗-algebra, which has locally finite decomposition rank, real rank zero and which absorbs the...
متن کاملA Universal Property for the Jiang-su Algebra
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 ...
متن کاملAsymptotically Unitary Equivalence and Classification of Simple Amenable C∗-algebras
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 φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(c)ut = φ2(c) for all c ∈ C if and only if [φ1] = [φ2] in KK(C,A), φ ‡ 1 = φ 2 ...
متن کاملThe Cuntz Semigroup of Continuous Functions into Certain Simple C∗-algebras
This paper contains computations of the Cuntz semigroup of separable C∗-algebras of the form C0(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...
متن کاملThe Classification of Simple Separable Unital Locally Ash Algebras
Let A be a simple separable unital locally approximately subhomogeneous C*algebra (locally ASH algebra). It is shown that A ⊗ Q can be tracially approximated by unital Elliott-Thomsen algebras with trivial K1-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.
متن کامل