We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*-algebra in its ultrapower depend on the choice of the ultrafilter? All algebras and all subalgebras in this note are C*-algebras and C*subalgebras, respectively, and all ultrafilters are nonprincipal ultrafilters on N. Our C*-terminology is standard (see e.g., [2]). In the following U ra...

Let X be a second countable, path connected, compact metric space and let A be a unital separable simple exact Z-stable real rank zero C∗-algebra. We classify all the embeddings (up to approximate unitary equivalence) of C(X) into A. Specifically, we prove the following: Theorem: Let α ∈ KL(C(X), A)+,1 and let λ : T (A) → T (C(X)) be an affine continuous map such that (i) if h ∈ Aff(T (C(X))) i...

We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C∗-algebra D; we also explain how the known classification theorems for nuclear C∗-algebras fit into this concept. As a new result in this direction, we show that the class of separable, unital, simple C∗-algebras with locally finite decomposition rank and UCT, and for which projections sep...

We show that any non-type I separable unital AF algebra B can be modeled from inside by a nonnuclear C*-algebra and from outside by a nonexact C*-algebra. More precisely there exist unital separable quasidiagonal C*-algebras A B C of real rank zero, stable rank one, such that A is nonnuclear, C is nonexact, and both A and C are asymptotically homotopy equivalent to B. In particular A, B and C h...

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two.

The rank of the adjacency matrix of a graph is bounded above by the number of distinct non-zero rows of that matrix. In general, the rank is lower than this number because there may be some non-trivial linear combination of the rows equal to zero. We show the somewhat surprising result that this never occurs for the class of cographs. Therefore, the rank of a cograph is equal to the number of d...

We introduce an order structure on K 0 ( ) 9 K0(; Z/p ) . This group may also be thought of as Ko(; 7z @ Z/p ) . We exhibit new examples of real-rank zero C*-algebras that are inductive limits of finite dimensional and dimension-drop algebras, have the same ordered, graded K-theory with order unit and yet are not isomorphic. In fact they are not even stably shape equivalent. The order structure...

We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus T as an ind-object in the category of holomorphic vector bundles on T . We define the rank of a quasicoherent sheaf that can take arbitrary nonnegative real values. We study the category Qcoh(ηT ) obtained by taking the quotient of the category of quasicoherent sheaves by the subcategory of objects of rank ze...

We find a distributive (∨, 0, 1)-semilattice Sω1 of size א1 that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: — There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with ...