Related papers: On $C^*$-norms on $\mathbb{Z}_2$-graded tensor pro…

We prove that a discrete group $G$ is amenable iff it is strongly unitarizable in the following sense: every unitarizable representation $\pi$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the…

For any unital separable simple infinite-dimensional nuclear C*-algebra with finitely many extremal traces, we prove that Z-absorption, strict comparison, and property (SI) are equivalent. We also show that any unital separable simple…

We show that a $C^*$-algebra $A$ is nuclear iff there is a constant $K$ and $\alpha<3$ such that, for any bounded homomorphism $u\colon A \to B(H)$, there is an isomorphism $\xi\colon H\to H$ satisfying $\|\xi^{-1}\|\|\xi\| \le…

We investigate the interplay of the following regularity properties for non-simple C*-algebras: finite nuclear dimension, Z-stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic…

Order unit property of a positive element in a $C^{*}$-algebra is defined. It is proved that precisely projections satisfy this order theoretic property. This way, unital hereditary $C^{*}$-subalgebras of a $C^{*}$-algebra are…

We present a classification theorem for amenable simple stably projectionless C*-algebras with generalized tracial rank one whose $K_0$ vanish on traces which satisfy the Universal Coefficient Theorem. One of them is denoted by ${\cal Z}_0$…

We prove a classification theorem for purely infinte simple C*-algebras that is strong enough to show that the tensor products of two different irrational rotation algebras with the same even Cuntz algebra are isomorphic. In more detail,…

We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C*-algebras. The concept is particularly relevant for Elliott's program to classify nuclear C*-algebras by K-theory…

Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this…

For a simple $C^*$-algebra $A$ and any other $C^*$-algebra $B$, it is proved that every closed ideal of $A \otimes^{\min} B$ is a product ideal if either $A$ is exact or $B$ is nuclear. Closed commutator of a closed ideal in a Banach…

We give two characterizations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for…

We present a new method, involving monads and comonads from category theory, to help establish a certain type of equivalence of subcategories. As a case study we consider the category of topological gradings of $C^*$-algebras over a fixed…

The Banach $^{*}$-algebra $A\hat{\otimes}B$, the operator space projective tensor product of $C^{*}$-algebras $A$ and $B$, is shown to be $^{*}$-regular if Tomiyama's property ($F$) holds for $A\otimes_{\min}B$ and $A \otimes_{\min}B=A…

The problem of expressing a selfadjoint element that is zero on every bounded trace as a finite sum (or a limit of sums) of commutators is investigated in the setting of C*-algebras of finite nuclear dimension. Upper bounds -- in terms of…

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the…

We show that separable, simple, unital C*-algebras with finite decomposition rank absorb the Jiang-Su algebra Z tensorially. This has a number of consequences for Elliott's program to classify nuclear C*-algebras by their K-theory data. In…

We prove that C*-algebras which, as Banach spaces, are Grothendieck cannot be decomposed into a tensor product of two infinite-dimensional C*-algebras. By a result of Pfitzner, this class contains all von Neumann algebras and their…

We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair and the space of quantum strategies…

The paper deals with $C^*$-algebras generated by a net of Hilbert spaces over a partially ordered set. The family of those algebras constitutes a net of $C^*$-algebras over the same set. It is shown that every such an algebra is graded by…

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…