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Boundaries of reduced C*-algebras of discrete groups
For a discrete group G, we consider the minimal C*-subalgebra of $\ell^\infty(G)$ that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up
C*-simplicity and the unique trace property for discrete groups
A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical
The Choquet boundary of an operator system
We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary representations to generate the C*-envelope.
The structure of an isometric tuple
An n‐tuple of operators (V1, …, Vn) acting on a Hilbert space H is said to be isometric if the operator [V1⋯Vn]:Hn→H is an isometry. We prove a decomposition for an isometric tuple of operators that
Essential normality and the decomposability of algebraic varieties
We consider the Arveson-Douglas conjecture on the essential normality of homogeneous submodules corresponding to algebraic subvarieties of the unit ball. We prove that the property of essential
Reduced twisted crossed products over C*-simple groups
We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and
Noncommutative Choquet theory
We introduce a new and extensive theory of noncommutative convexity along with a corresponding theory of noncommutative functions. We establish noncommutative analogues of the fundamental results
Wandering vectors and the reflexivity of free semigroup algebras
Abstract A free semigroup algebra is the weak-operator-closed (non-self-adjoint) operator algebra generated by n isometries with pairwise orthogonal ranges. A unit vector x is said to be wandering
The classification problem for finitely generated operator systems and spaces
The classification of separable operator systems and spaces is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm
Characterizations of C*-simplicity
We characterize discrete groups that are C*-simple, meaning that their reduced C*-algebra is simple. First, we prove that a discrete group is C*-simple if and only if it has Powers' averaging