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Register a new userStrongly peaking representations and compressions of operator systems
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We use Arvesons notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets which admit minimal presentations. A fully compressed separable operator system necessarily generates the C*-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.
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We study the matrix range of a tuple of compact operators on a Hilbert space and examine the notions of minimal, nonsingular, and fully compressed tuples. In this pursuit, we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range. Further, we find that a compact tuple $A$ is fully compressed if and only if it is multiplicity-free and the Shilov ideal is trivial, which occurs if and only if $A$ is minimal and nonsingular. Fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges. We also produce a proof of this fact which does not depend on the concept of nonsingularity.
We establish the dual equivalence of the category of (potentially nonunital) operator systems and the category of pointed compact nc (noncommutative) convex sets, extending a result of Davidson and the first author. We then apply this dual equivalence to establish a number of results about operator systems, some of which are new even in the unital setting. For example, we show that the maximal and minimal C*-covers of an operator system can be realized in terms of the C*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We also characterize C*-simple operator systems, i.e. operator systems with simple minimal C*-cover, in terms of their nc quasistate spaces. We develop a theory of quotients of operator systems that extends the theory of quotients of unital operator algebras. In addition, we extend results of the first author and Shamovich relating to nc Choquet simplices. We show that an operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdans property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra.
We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselters generalization of the Cuntz-Pimsner C*-algebra to the context of subproduct systems. Suppose that $X$ and $Y$ are Arveson-Stinespring subproduct systems associated to two stochastic matrices over a countable set $Omega$, and let $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ be their tensor algebras. We show that every algebraic isomorphism from $mathcal{T}_+(X)$ onto $mathcal{T}_+(Y)$ is automatically bounded. Furthermore, $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ are isometrically isomorphic if and only if $X$ and $Y$ are unitarily isomorphic up to a *-automorphism of $ell^infty(Omega)$. When $Omega$ is finite, we prove that $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ are algebraically isomorphic if and only if there exists a similarity between $X$ and $Y$ up to a *-automorphism of $ell^infty(Omega)$. Moreover, we provide an explicit description of the Cuntz-Pimsner algebra $mathcal{O}(X)$ in the case where $Omega$ is finite and the stochastic matrix is essential.
We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Aglers theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.
Let $A$ be a unital operator algebra and let $alpha$ be an automorphism of $A$ that extends to a *-automorphism of its $ca$-envelope $cenv (A)$. In this paper we introduce the isometric semicrossed product $A times_{alpha}^{is} bbZ^+ $ and we show that $cenv(A times_{alpha}^{is} bbZ^+) simeq cenv (A) times_{alpha} bbZ$. In contrast, the $ca$-envelope of the familiar contractive semicrossed product $A times_{alpha} bbZ^+ $ may not equal $cenv (A) times_{alpha} bbZ$. Our main tool for calculating $ca$-envelopes for semicrossed products is the concept of a relative semicrossed product of an operator algebra, which we explore in the more general context of injective endomorphisms. As an application, we extend a recent result of Davidson and Katsoulis to tensor algebras of $ca$-correspondences. We show that if $T_{X}^{+}$ is the tensor algebra of a $ca$-correspondence $(X, fA)$ and $alpha$ a completely isometric automorphism of $T_{X}^{+}$ that fixes the diagonal elementwise, then the contractive semicrossed product satisfies $ cenv(T_{X}^{+} times_{alpha} bbZ^+)simeq O_{X} times_{alpha} bbZ$, where $O_{X}$ denotes the Cuntz-Pimsner algebra of $(X, fA)$.
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