We expand upon work from many hands on the decomposition of nuclear maps. Such maps can be characterised by their ability to be approximately written as the composition of maps to and from matrices. Under certain conditions (such as quasidiagonality), we can find a decomposition whose maps behave nicely, by preserving multiplication up to an arbitrary degree of accuracy and being constructed from order-zero maps (as in the definition of nuclear dimension). We investigate these conditions and relate them to a W*-analogue.

Main result: If a ${{C}^{*}}$-algebra $\mathcal{A}$ is simple, $\sigma $-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra $\mathcal{M}\left( \mathcal{A} \right)$ also has strict comparison of positive elements by traces. The same results holds if finitely many extremal traces is replaced by quasicontinuous scale. A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma $-unital ${{C}^{*}}$ -algebra can be approximated by a bi-diagonal series. As an application of strict comparison, if $\mathcal{A}$ is a simple separable stable ${{C}^{*}}$ -algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

Let A = C(X) ⊗ K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space. Let A s be the algebra of strong*-continuous functions from X to K(H). Then A s /A is the inner corona algebra of A. We show that if X has no isolated points, then A s /A is an essential ideal of the corona algebra of A, and Prim(A s /A), the primitive ideal space of A s /A, is not weakly Lindelof. If X is also first countable, then there is a natural injection from the power set of X to the lattice of closed ideals of A s /A. If X = βℕ\ℕ and the continuum hypothesis (CH) is assumed, then the corona algebra of A is a proper subalgebra of the multiplier algebra of A s /A. Several of the results are obtained in the more general setting of C 0(X)-algebras.

We offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.

The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple ${{C}^{*}}$-algebras arising from free product constructions, Canad. J. Math. 50 (1998), 323–341 and of Theorem 3.1 of K J. Dykema, Purely infinite simple ${{C}^{*}}$-algebras arising from free product constructions, II, Math. Scand. 90 (2002), 73–86 are corrected.

In this paper we consider near inclusions $A\,{{\subseteq }_{\gamma }}\,B$ of ${{\text{C}}^{*}}$-algebras. We show that if $B$ is a separable type $\text{I}$${{\text{C}}^{*}}$-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type $\text{I}$. We then use this to obtain an embedding of $A$ into $B$.

Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and $K$-groups of our algebras.

Examples of simple, separable, unital, purely infinite ${{C}^{*}}$-algebras are constructed, including:

1. (1) some that are not approximately divisible;

2. (2) those that arise as crossed products of any of a certain class of ${{C}^{*}}$-algebras by any of a certain class of non–unital endomorphisms;

3. (3) those that arise as reduced free products of pairs of ${{C}^{*}}$-algebras with respect to any from a certain class of states.