We study sufficient conditions under which a nowhere scattered $\mathrm {C}^*$-algebra $A$ has a nowhere scattered multiplier algebra $\mathcal {M}(A)$, that is, we study when $\mathcal {M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $\sigma$-unital $\mathrm {C}^*$-algebra $A$ of

1. (i) finite nuclear dimension, or

2. (ii) real rank zero, or

3. (iii) stable rank one with $k$-comparison,

$\mathcal {M}(A)$

In this paper, let A be an infinite-dimensional stably finite unital simple separable $\mathrm {C^*}$ -algebra. Let $B\subset A$ be a centrally large subalgebra in A such that B has uniform property $\Gamma $ . Then we prove that A has uniform property $\Gamma $ . Let $\Omega $ be a class of stably finite unital $\mathrm {C^*}$ -algebras such that for any $B\in \Omega $ , B has uniform property $\Gamma $ . Then we show that A has uniform property $\Gamma $ for any simple unital $\mathrm {C^*}$ -algebra $A\in \rm {TA}\Omega $ .

We study ring-theoretic (in)finiteness properties—such as Dedekind-finiteness and proper infiniteness—of ultraproducts (and more generally, reduced products) of Banach algebras.

While we characterise when an ultraproduct has these ring-theoretic properties in terms of its underlying sequence of algebras, we find that, contrary to the $C^*$ -algebraic setting, it is not true in general that an ultraproduct has a ring-theoretic finiteness property if and only if “ultrafilter many” of the underlying sequence of algebras have the same property. This might appear to violate the continuous model theoretic counterpart of Łoś’s Theorem; the reason it does not is that for a general Banach algebra, the ring theoretic properties we consider cannot be verified by considering a bounded subset of the algebra of fixed bound. For Banach algebras, we construct counter-examples to show, for example, that each component Banach algebra can fail to be Dedekind-finite while the ultraproduct is Dedekind-finite, and we explain why such a counter-example is not possible for $C^*$ -algebras. Finally, the related notion of having stable rank one is also studied for ultraproducts.

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.

We provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C*-algebras. Its functoriality properties are explored, and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the one recently proposed by Gardella [‘Regularity properties and Rokhlin dimension for compact group actions’, Houston J. Math.43(3) (2017), 861–889], and we complement their work by providing an open projection picture of it.

We study comparison properties in the category $\text{Cu}$ aiming to lift results to the ${{\text{C}}^{\text{*}}}$-algebraic setting. We introduce a new comparison property and relate it to both the corona factorization property $\left( \text{CFP} \right)$ and $\omega $-comparison. We show differences of all properties by providing examples that suggest that the corona factorization for ${{\text{C}}^{\text{*}}}$-algebras might allow for both finite and infinite projections. In addition, we show that Rørdam's simple, nuclear ${{\text{C}}^{\text{*}}}$-algebra with a finite and an inifnite projection does not have the $\text{CFP}$.

We examine the ranks of operators in semi-finite ${{C}^{*}}$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple ${{C}^{*}}$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray–von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first author, this shows that slow dimension growth coincides with $Z$-stability for approximately subhomogeneous algebras.

Let $X$ be a compact metric space. A lower bound for the radius of comparison of the ${{\text{C}}^{*}}$-algebra $\text{C}\left( X \right)$ is given in terms of ${{\dim}_{\mathbb{Q}}}\,X$, where ${{\dim}_{\mathbb{Q}}}\,X$ is the cohomological dimension with rational coefficients. If ${{\dim}_{\mathbb{Q}}}\,X\,=\,\dim\,X\,=\,d$, then the radius of comparison of the ${{\text{C}}^{*}}$-algebra $C\left( X \right)$ is $\max \left\{ 0,\,\left( d\,-\,1 \right)/\,2\,-\,1 \right\}$ if $d$ is odd, and must be either ${d}/{2}\;\,-\,1$ or ${d}/{2}\;\,-\,2$ if $d$ is even (the possibility ${d}/{2}\;\,-\,1$ does occur, but we do not know if the possibility ${d}/{2}\;\,-\,2$ can also occur).