We prove that the monoidal 2-category of cospans of ordinals and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2-dimensional separable algebra condition.

We classify all essential extensions of the form 0 → B → D → C(X) → 0 where B is a nonunital simple separable finite real rank zero Z-stable C*algebra with continuous scale, and where X is a finite CW complex. In fact, we prove that there is a group isomorphism Ext(C(X),B) → KK(C(X),M(B)/B).

The Connes Embedding Problem (CEP) asks whether every separable II1 factor embeds into an ultrapower of the hyperfinite II1 factor. We show that the CEP is equivalent to the statement that every type II1 tracial von Neumann algebra has a computable universal theory.

Let A be a separable algebra (with an involution). The varieties of flags of (isotropic) ideals of A are considered and certain decompositions of these varieties in the category of Chow-correspondences are produced. As a consequence, decompositions in various cohomology theories are obtained.

Kasparov KK-groups KK(A,B) are represented as homotopy groups of the Pedersen-Weibel nonconnective algebraic K-theory spectrum of the additive category of Fredholm (A,B)-bimodules for A and B, respectively, a separable and σ-unital trivially graded real or complex C∗-algebra acted upon by a fixed compact metrizable group.

We prove that the monoidal 2-category of cospans of finite linear orders and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2-dimensional separable algebra condition.

Let (A, α) and (B, β) be C*-dynamical systems and assume that A is a separable simple C*-algebra and that α and β are ∗-automorphisms. Then the semicrossed products A×αZ and B ×β Z are isometrically isomorphic if and only if the dynamical systems (A, α) and (B, β) are outer conjugate.

We show that separable, simple, nonelementary, unital C∗-algebras with finite decomposition rank absorb the Jiang–Su algebra Z tensorially. This has a number of consequences for Elliott’s program to classify nuclear C∗algebras by their K-theory data. In particular, it completes the classification of C∗-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K...

We have seen that the proof of existence of inverses for elements of Ext(X) can be based on a lifting theorem for (completely) positive maps of C(X) into a quotient C∗-algebra of the form E/K, where E ⊆ B(H) is a C∗-algebra containing the compact operators K. That argument works equally well for arbitrary C∗-algebras in place of C(X) whenever a completely positive lifting exists. Thus we are le...

Let M be a smooth Fredholm manifold modeled on a separable infinite-dimensional Euclidean space E with Riemannian metric g. Given an augmented Fredholm filtration F of M by finite-dimensional submanifolds {Mn}∞n=k, we associate to the triple (M, g,F) a non-commutative direct limit C-algebra A(M, g,F) = lim −→ A(Mn) that can play the role of the algebra of functions vanishing at infinity on the ...