Related papers: Non-linear positive maps between $C^*$-algebras

We say a completely positive contractive map between two C*-algebras has order zero, if it sends orthogonal elements to orthogonal elements. We prove a structure theorem for such maps. As a consequence, order zero maps are in one-to-one…

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,…

When a unital \ca $A$ has topological stable rank one (write $\tsr(A) = 1$), we know that $\tsr(pAp) \leq 1$ for a non-zero projection $p \in A$. When, however, $\tsr(A) \geq 2$, it is generally faluse. We prove that if a unital C*-algebra…

Let $\mathcal A$ and $\mathcal B$ be two (complex) algebras. A linear map $\phi:{\mathcal A}\to{\mathcal B}$ is called $n$-homomorphism if $\phi(a_{1}... a_{n})=\phi(a_{1})...\phi(a_{n})$ for each $a_{1},...,a_{n}\in{\mathcal A}.$ In this…

We investigate linear operators between C$^\ast$-algebras which approximately preserve involution and orthogonality, the latter meaning that for some $\varepsilon>0$ we have $\|\phi(x)\phi(y)\|\leq\varepsilon\|x\|\|y\|$ for all positive…

We establish necessary and sufficient conditions on a (not necessarily countable) graph E for the graph C*-algebra C*(E) to be primitive. Along with a known characterization of the graphs E for which C*(E) is prime, our main result provides…

We define variants of Pisier's similarity degree for unital C*-algebras and use direct integral theory to obtain new results. We prove that if every II$_{1}$ factor representation of a separable C*-algebra $\mathcal{A}$ has property…

We study nonunital $C^*$-algebras such that for any element there exists a local unit and prove that in such algebras there are no frames. This fact was previously known only for commutative algebras. Among other results, we establish some…

A map $\Phi$ between matrices is said to be zero product preserving if $$ \Phi(A)\Phi(B) = 0 \quad \text{whenever}\quad AB = 0. $$ In this paper, we give concrete descriptions of an additive/linear zero product preserver $\Phi: {\bf…

We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for \alpha. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show…

A $C^*$-algebra $A$ is said to have the homotopy lifting property if for all $C^*$-algebras $B$ and $E$, for every surjective $^*$-homomorphism $\pi \colon E \rightarrow B$ and for every $^*$-homomorphism $\phi \colon A \rightarrow E$, any…

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…

We consider two inclusions of $C^*$-algebras whose small $C^*$-algebras have approximate units of the large $C^*$-algebras and their two spaces of all bounded bimodule linear maps. We suppose that the two inclusions of $C^*$-algebras are…

Let $A$ be a unital $C^*$-algebra and let $U_0(A)$ be the group of unitaries of $A$ which are path connected to the identity. Denote by $CU(A)$ the closure of the commutator subgroup of $U_0(A).$ Let $i_A^{(1, n)}\colon…

We identify and characterize unital completely positive (UCP) maps on finite dimensional $C^*$-algebras for which the Choi-Effros product extended to the space generated by peripheral eigenvectors matches with the original product. We…

The problem of inner vs outer conjugacy of subalgebras of certain graph C*-algebras is investigated. For a large class of finite graphs E, we show that whenever $\alpha$ is a vertex-fixing quasi-free automorphism of the corresponding graph…

It is known that a graph $C^*$-algebra $C^*(E)$ is approximately finite dimensional (AF) if and only if the graph $E$ has no loops. In this paper we consider the question of when a labeled graph $C^*$-algebra $C^*(E,\CL,\CB)$ is AF. A…

We show that a simple separable unital nuclear nonelementary $C^*$-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large order zero maps from matrix algebras into…

An oriented graph is said positively multiplicative when its adjacency matrix $A$ embeds in a matrix algebra admitting a basis $\mathsf{B}$ with nonnegative structure constants in which the matrix of the multiplication by $A$ coincides with…

Let A be a commutative unital C*-algebra and let S denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of A to be unitarily equivalent to a multiplicative representation on a…