We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and algebras, commutative $C^*$-algebras von Neumann algebras equipped with traces, in “uncountable” setting wh

C -algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C -algebras. At first, this paper concentrates on the com...

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 ...

The usual crossed product construction which associates to the homeomorphism T of the locally compact space X the C∗-algebra C∗(X, T ) is extended to the case of a partial local homeomorphism T . For example, the Cuntz-Krieger algebras are the C∗-algebras of the one-sided Markov shifts. The generalizations of the Cuntz-Krieger algebras (graph algebras, algebras OA where A is an infinite matrix)...

In this paper we show that a strongly homotopy commutative (or C∞-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞-algebra (an ∞-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a C∞-algebra and does not generalize to a...

The noncommutative analog of an approximative absolute retract (AAR) is introduced, a weakly projective C∗-algebra. This property sits between being residually finite dimensional and projectivity. Examples and closure properties are considered.

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra A is a ∗-homomorphism A → M that factors through the canonical inclusion C(X) ⊆ `∞(X) when restricted to a commutative C*-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as well as a possibly noninjective functorial discretization, where M is a C*...

It is proved that classifiable simple separable nuclear purely infinite C∗algebras having finitely generated K-theory and torsion-free K1 are semiprojective. This is accomplished by exhibiting these algebras as C∗-algebras of infinite directed graphs.

In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants.