Related papers: Cohomological obstructions to lifting properties f…

We characterize the lifting property (LP) of a separable $C^*$-algebra $A$ by a property of its maximal tensor product with other $C^*$-algebras, namely we prove that $A$ has the LP if and only if for any family $(\{D_i\mid i\in I\}$ of…

The Local Lifting Property (LLP) is a localized version of projectivity for completely positive maps between $\mathrm{C}^*$-algebras. Outside of the nuclear case, very few $\mathrm{C}^*$-algebras are known to have the LLP. In this article,…

In this paper, we employ operator system techniques to investigate structural properties of C*-algebras. In particular, we provide more direct proofs of results concerning exactness and the local lifting property (LLP) of group…

We give several simple and easy complements to our recent paper on $C^*$-algebras with the lifting property (LP in short). In particular we observe that the local lifting property (LLP in short) associated to the class of max-contractions…

This note is motivated by Kirchberg's conjecture that the local lifting property (LLP) implies the lifting property (LP) for $C^*$-algebras. The author recently constructed by a "local" method an example of $C^*$-algebra with the LLP and…

We construct the first example of a $C^*$-algebra $A$ with the properties in the title. This gives a new example of non-nuclear $A$ for which there is a unique $C^*$-norm on $A \otimes A^{op}$. This example is of particular interest in…

The (Local) Lifting Property ((L)LP) is introduced by Kirchberg and deals with lifting completely positive maps. We give a characterization of the (L)LP in terms of lifting $\ast$-homomorphisms. We use it to prove that if $A$ and $B$ have…

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…

Let $\Gamma$ be a discrete group. A $C^*$-algebra $A$ is an exotic $C^*$-algebra (associated to $\Gamma$) if there exist proper surjective $C^*$-quotients $C^*(\Gamma)\to A\to C^*_r(\Gamma)$. In this paper, we show that a large class of…

We study the Haagerup property for C*-algebras. We first give new examples of C*-algebras with the Haagerup property. A nuclear C*-algebra with a faithful tracial state always has the Haagerup property, and the permanence of the Haagerup…

We give a new, somewhat simpler, presentation of the author's recent construction of a non-nuclear $C^*$-algebra which has both the local lifting property (LLP) and the weak expectation property (WEP).

We show that for a Hecke pair $(G, \Gamma)$ the $C^*$-completions $C^*(L^1(G, \Gamma))$ and $pC^*(\bar{G})p$ of its Hecke algebra coincide whenever the group algebra $L^1(\bar{G})$ satisfies a spectral property which we call…

Given a short exact sequence of locally compact abelian groups $0 \to A \to B \to C \to 0$ and a continuous $C$-valued $1$-cocycle $\phi$ on a locally compact Hausdorff groupoid $\Gamma$ we construct a twist of $\Gamma$ by $A$ that is…

Let $\Gamma$ be a countable discrete group. We say that $\Gamma$ has $C^*$-invariant subalgebra rigidity (ISR) property if every $\Gamma$-invariant $C^*$-subalgebra $\mathcal{A}\le C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for some normal…

For a locally convex $^*$-algebra $A$ equipped with a fixed continuous $^*$-character $\varepsilon$, we define a cohomological property, called property $(FH)$, which is similar to character amenability. Let $C_c(G)$ be the space of…

Approximate morphisms have seen significant study across many areas of mathematics, for instance, in the theory of Absolute (Neighborhood) Retracts in topology, or of almost-commuting unitary matrices in analysis. This paper initiates study…

The following homotopy lifting theorem is proved: Let $\phi, \psi: B \to D/I$ be homotopic $\ast$-homomorphisms and suppose $\psi$ lifts to a (discrete) asymptotic homomorphism. Then $\phi$ lifts to a (discrete) asymptotic homomorphism.…

A host algebra of a topological group G is a C^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite…

We study lifting properties for full product C*-algebras with amalgamation over ${\mathbb C}1$ and give new proofs for some results of Kirchberg and Pisier. We extend the result of Choi on the quasidiagonality of $C^*({\mathbb F}_n)$,…

Let $G$ be a locally compact group and let $C^*(G)$ and $C^*_r(G)$ be the full group $C^*$-algebra and the reduced group $C^*$-algebra of $G$. We investigate the relationship between Property $(T)$ for $G$ and Property $(T)$ as well as its…