Related papers: Nowhere scattered multiplier algebras

We say that a C*-algebra is nowhere scattered if none of its quotients contains a minimal open projection. We characterize this property in various ways, by topological properties of the spectrum, by divisibility properties in the Cuntz…

Let $A$ be a $C^*$-algebra. It is shown that the following conditions are equinvalent: (1) $A$ is scattered, (2) every $C^*$-subalgebra of $A$ is AF, (3) every $C^*$-subalgebra of $A$ has real rank zero.

In this paper, we introduce the notion of a scattered locally $C^{\ast }$ -algebra and we give conditions for a locally $C^{\ast }$-algebra to be scattered. Given an action $\alpha $ of a locally compact group $G$ on a scattered locally…

We show that a C*-algebra with topological dimension zero has the Global Glimm Property (every hereditary subalgebra contains an almost full nilpotent element) if and only if it is nowhere scattered (no hereditary subalgebra admits a…

In this paper we investigate the extremal richness of the multiplier algebra $M(A)$ and the corona algebra $M(A)/A$, for a simple C*-algebra $A$ with real rank zero and stable rank one. We show that the space of extremal quasitraces and the…

It is shown that if a C*-algebra has nuclear dimension $n$ then its Cuntz semigroup has the property of $n$-comparison. It then follows from results by Ortega, Perera, and Rordam that $\sigma$-unital C*-algebras of finite nuclear dimension…

We introduce a concept of the bounded rank (with respect to a positive constant) for unital C*-algebras as a modification of the usual real rank and present a series of conditions insuring that bounded and real ranks coincide. These…

Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $\theta : E\to F$ is a linear map preserving orthogonality, i.e., $<\theta(x), \theta(y) > = 0$ whenever $<x, y > = 0$. We show in this…

A well-known theorem of Blackadar and Handelman states that every unital stably finite C*-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the…

Suppose that A is a C*-algebra for which A is isomorphic to A tensor Z, where Z is the Jiang-Su algebra: a unital, simple, stably finite, separable, nuclear, infinite dimensional C*-algebra with the same Elliott invariant as the complex…

It is known that a C*-algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by…

We introduce the completely positive rank, a notion of covering dimension for nuclear $C^*$-algebras and analyze some of its properties. The completely positive rank behaves nicely with respect to direct sums, quotients, ideals and…

We show that there exists a separable, nuclear C*-algebra with real rank zero and trivial K-theory such that its multiplier and corona algebra have real rank one. This disproves two conjectures of Brown and Pedersen. We also compute the…

Let X be a path connected, compact metric space and let A be a unital separable simple nuclear Z-stable real rank zero C*-algebra. We classify all the unital *-embeddings (up to approximate unitary equivalence) of C(X) into A. Specifically,…

We show that for a large class of C*-algebras $\mathcal{A}$, containing arbitrary direct limits of separable type I C*-algebras, the following statement holds: If $A\in \mathcal{A}$ and $B$ is a simple projectionless C*-algebra with trivial…

Let $A$ be an algebra over any field. We do not assume that $A$ has an identity. The \emph{multiplier algebra} $M(A)$ is a unital algebra associated to $A$. If we require the product in $A$ to be non-degenerate (as a bilinear form), the…

For elements $a, b$ of a C*-algebra we denote $a=ab$ by $a\ll b$. We show that all $\omega_1$-unital C*-algebras have $\ll$-increasing approximate units, extending a classical result for $\sigma$-unital C*-algebras. We also construct (in…

In this paper, we give two properties of C*-algebra that could be deduced from the properties of its large subalgebra. Let A be an infinite dimensional simple unital C*-algebra and let B be a centrally large subalgebra of A, we prove that A…

Main result: If a C*-algebra is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier also has strict comparison of positive elements by traces. The same…

Given a closed ideal $I$ in a C*-algebra $A$, we show that $A$ is pure if and only if $I$ and $A/I$ are pure. More generally, we study permanence of comparison and divisibility properties when passing to extensions. As an application we…