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Given an operator system S$\mathcal {S}$, we define the parameters rk(S)$r_k(\mathcal {S})$ (resp. dk(S)$d_k(\mathcal {S})$) defined as the maximal value of the completely bounded norm of a unital k$k$‐positive map from an arbitrary operator system into S$\mathcal {S}$ (resp. from S$\mathcal {S}$ into an arbitrary operator system). In the case of the matrix algebras Mn$\mathsf {M}_n$, for 1⩽k⩽n$1 \leqslant k \leqslant n$, we compute the exact value rk(Mn)=2n−kk$r_k(\mathsf {M}_n) = \frac{2n-k}{k}$ and show upper and lower bounds on the parameters dk(Mn)$d_k(\mathsf {M}_n)$. Moreover, when S$\mathcal {S}$ is a finite‐dimensional operator system, adapting results of Passer and the fourth author [J. Operator Theory 85 (2021), no. 2, 547–568], we show that the sequence (rk(S))$(r_k(\mathcal {S}))$ tends to 1 if and only if S$\mathcal {S}$ is exact and that the sequence (dk(S))$(d_k(\mathcal {S}))$ tends to 1 if and only if S$\mathcal {S}$ has the lifting property. [ABSTRACT FROM AUTHOR]

The theory of positive maps plays a central role in operator algebras and functional analysis and has countless applications in quantum information science. The theory was originally developed for operators acting on complex Hilbert spaces and less is known about its variant on real Hilbert spaces. In this article, we study positive maps acting on a full matrix algebra over the reals, pointing out a number of fundamental differences with the complex case and discussing their implications in quantum information. We provide a necessary and sufficient condition for a real map to admit a positive complexification and connect the existence of positive maps with non-positive complexification with the existence of mixed states that are entangled in real Hilbert space quantum mechanics, but separable in the complex version, providing explicit examples both for the maps and for the states. Finally, we discuss entanglement breaking and PPT maps, and we show that a straightforward real version of the PPT-squared conjecture is false even in dimension 2. Nevertheless, we show that the original PPT-squared conjecture implies a different conjecture for real maps, in which the PPT property is replaced by a stronger property of invariance under partial transposition (IPT). When the IPT property is assumed, we prove an asymptotic version of the conjecture. [ABSTRACT FROM AUTHOR]

We study the multiplier algebras A.H/obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces H on the ball Bd of Cd. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of A.H/in terms of the complementary bands of Henkin and totally singular measures for Mult.H/. This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact Mult.H/-totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is Mult.H/-totally null. [ABSTRACT FROM AUTHOR]

Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space. [ABSTRACT FROM AUTHOR]

We use Arveson's notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets that 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. [ABSTRACT FROM AUTHOR]

A free semigroupoid algebra is the wot -closure of the algebra generated by a TCK family of a graph. We obtain a structure theory for these algebras analogous to that of free semigroup algebra. We clarify the role of absolute continuity and wandering vectors. These results are applied to obtain a Lebesgue-von Neumann-Wold decomposition of TCK families, along with reflexivity, a Kaplansky density theorem and classification for free semigroupoid algebras. Several classes of examples are discussed and developed, including self-adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence. [ABSTRACT FROM AUTHOR]

We define the complete numerical radius norm for homomorphisms from any operator algebra into B(H), and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if K is a complete C-spectral set for an operator T, then it is a complete M-numerical radius set, where M = 1/2 (C + C-1). In particular, in view of Crouzeix's theorem, there is a universal constant M (less than 5.6) so that if P is a matrix polynomial and T ∈ B(H), then w(P(T)) ≤ || P || W(T). When W(T) = D, we have M = 5/4. [ABSTRACT FROM AUTHOR]

We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-* closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments. [ABSTRACT FROM AUTHOR]

Semicrossed product algebras have been used to study dy-namical systems since their introduction by Arveson in 1967. In this survey article, we discuss the history and some recent work, focussing on the conjugacy problem, dilation theory and C*-envelopes, and some connections back to the dynamics. [ABSTRACT FROM AUTHOR]

A matrix convex set is a set of the form S = Un≥1 Sn (where each Sn is a set of d-tuples of n x n matrices) that is invariant under unital completely positive maps from Mn to Mk and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough, and Schweighofer. Given two matrix convex sets S = Un≥1 Sn, and T = Un≥1 Tn, we find geometric conditions on S or on T, such that S1 ⊆ T1 implies that S ⊆ CT for some constant C. For instance, under various symmetry conditions on S, we can show that C above can be chosen to equal d, the number of variables.We also show that C = d is sharp for a specific matrix convex set Wmax (...) constructed from the unit ball Bd. This led us to find an essentially unique self-dual matrix convex set D, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant C = √d. For a certain class of polytopes, we obtain a considerable sharpening of such inclusion results involving polar duals. An illustrative example is that a sufficient condition for T to contain the free matrix cube C(d) = Un {(T1, ..., Td) ∈ Mnd : ||Ti||, is that {x ∈ Rd : Σ |xj| ≤ 1} ⊆ 1/d T1, that is, that 1/d T⊆ contains the polar dual of the cube [-1, 1]d = C1(d). Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the "number of variables" d. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators. [ABSTRACT FROM AUTHOR]

Absolutely continuous commuting row contractions admit a weak-⁎ continuous functional calculus. Building on recent work describing the first and second dual spaces of the closure of the polynomial multipliers on the Drury–Arveson space, we give a complete characterization of these commuting row contractions in measure theoretic terms. We also establish that completely non-unitary row contractions are necessarily absolutely continuous, in direct parallel with the case of a single contraction. Finally, we consider refinements of this question for row contractions that are annihilated by a given ideal. [ABSTRACT FROM AUTHOR]

We consider the closed algebra A d generated by the polynomial multipliers on the Drury–Arveson space. We identify A d ⁎ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop–Carleson–Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light on the nature of the extreme points of the unit ball of A d ⁎ . [ABSTRACT FROM AUTHOR]

We identify the exposed points of the unit ball of the dual space of the ball algebra. As a corollary, we show that the predual of H∝(Bd) has no extreme points in its unit ball. [ABSTRACT FROM AUTHOR]

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions MV of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball Bd. We find that MV is completely isometrically isomorphic to MW if and only if W is the image of V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when d < ∞ every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V and W are each a finite union of irreducible varieties and a discrete variety, when d < ∞, an isomorphism between MV and MW determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-* continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold-- particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences. [ABSTRACT FROM AUTHOR]

Let (A,α) and (B,β) be C*-dynamical systems where α and β are arbitrary *-endomorphisms. When α is injective or surjective, we show that the semicrossed products and are isometrically isomorphic if and only if (A,α) and (B,β) are outer conjugate. This conclusion also holds in various other cases as well. [ABSTRACT FROM PUBLISHER]

We define four 3×3 commuting contractions which do not dilate to commuting isometries. However, they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three 3×3 commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries. [ABSTRACT FROM PUBLISHER]

If α is the endomorphism of the disk algebra, A(D), induced by composition with a finite Blaschke product b, then the semicrossed product A(D) ×α Z+ imbeds canonically, completely isometrically into C(T) ×α Z+. Hence in the case of a non-constant Blaschke product b, the C*-envelope has the form C(Sb) ×s Z, where (Sb, s) is the solenoid system for (T, b). In the case where b is a constant, the C*-envelope of A(D) ×α Z+ is strongly Morita equivalent to a crossed product of the form C0(Se) ×s Z, where e: T × N -→ T × N is a suitable map and (Se, s) is the solenoid system for (T × N, e). [ABSTRACT FROM AUTHOR]

Abstract: This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The -envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well. [Copyright &y& Elsevier]

Abstract: Let φ be an isometric automorphism of the non-commutative disc algebra for . We show that every contractive covariant representation of dilates to a unitary covariant representation of . Hence the C∗-envelope of the semicrossed product is . [Copyright &y& Elsevier]

If T=[T1 … Tn] is a row contraction with commuting entries, and the Arveson dilation is T̃=[T̃1 … T̃n], then any operator X commuting with each Ti dilates to an operator Y of the same norm that commutes with each T̃i. [ABSTRACT FROM PUBLISHER]

Abstract: We provide a detailed analysis of atomic ∗-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal ∗-dilations of defect free representations modelled on finite groups of rank 2. [Copyright &y& Elsevier]

Abstract: The subspaces and subalgebras of which are hyperreflexive with constant 1 are completely classified. It is shown that there are 1-hyperreflexive subspaces for which the complete hyperreflexivity constant is strictly greater than 1. The constants for are analyzed in detail. [Copyright &y& Elsevier]

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra ${\mathfrak{L}}_G$ of a countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra ${\mathcal{T}}^{+}(G)$.We prove that the finite-dimensional nest representations separate the points in ${\mathfrak{L}}_G$, and a fortiori, in ${\mathcal{T}}^{+}(G)$. The irreducible finite-dimensional representations separate the points in ${\mathfrak{L}}_G$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every vertex $x \in {\mathcal{T}}(G)$ supporting a cycle, $x$ also supports at least one loop edge.We also study faithful nest representations. We prove that ${\mathfrak{L}}_G$ (or ${\mathcal{T}}^{+}(G)$) admits a faithful irreducible representation if and only if $G$ is strongly transitive as a directed graph. More generally, we obtain a condition on $G$ which is equivalent to the existence of a faithful nest representation. We also give a condition that determines the existence of a faithful nest representation for a maximal type ${\mathbb{N}}$ nest. [ABSTRACT FROM AUTHOR]

Abstract: Pati showed that every 4×4 matrix is unitarily similar to a tridiagonal matrix. We give a simple proof. In addition, we show that (in an appropriate sense) there are generically precisely 12 ways to do this. When the real part is diagonal, it is shown that the unitary can be chosen with the form U = PD where D is diagonal and P is real orthogonal. However even if both real and imaginary parts are real symmetric, there may be no real orthogonal matrices which tridiagonalize it. On the other hand, if the matrix belongs to the Lie algebra , then it can be tridiagonalized by a unitary in the symplectic group Sp(2). In dimension 5 or greater, there are always rank three matrices which are not tridiagonalizable. [Copyright &y& Elsevier]

Abstract: We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra . Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a -extendible representation . A -extendible representation of is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given -extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of is weak- dense in the unit ball of the associated free semigroup algebra if and only if is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts. [Copyright &y& Elsevier]

In this paper we give criteria for an ideal J of a TAF algebra A to be meet-irreducible. We show that J is meet-irreducible if and only if the C&ast;-envelope of A/J is primitive. In that case, A/J admits a faithful nest representation which extends to a &ast;-representation of the C&ast;-envelope for A/J. We also characterize the meet-irreducible ideals as the kernels of bounded nest representations; this settles the question of whether the n-primitive and meet-irreducible ideals coincide. [Copyright &y& Elsevier]

In this paper, we study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of an NSAF algebra by any closed ideal has an AF C*-envelope, and this algebra is exhibited as a quotient of a concretely represented AF-algebra. When the ideal is prime, the C*-envelope is primitive. The GNS construction is used to produce algebraically irreducible (in fact n-transitive for all n&ges;1) representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover, these representations extend to *-representations of the C*-envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the {0} ideal. [Copyright &y& Elsevier]

A free semigroup algebra is the weak operator topology closed algebra generated by a set of isometries with pairwise orthogonal ranges. The most important example is the left regular free semigroup algebra generated by the left regular representation of the free semigroup on $n$ generators. This algebra is the appropriate non-commutative $n$-dimensional analogue of the analytic Toeplitz algebra. We develop a detailed picture of the invariant subspace structure analogous to Beurling's theorem and show that this algebra is hyper-reflexive with distance constant at most 51.The free semigroup algebras, known as atomic, for which the range projections of words in the generators lie in an atomic masa are completely classified. This provides a complete classification for a large class of representations of the Cuntz C*-algebras $\mathcal{O}_n$. This allows us to describe completely the invariant subspace structure of these algebras, and thereby show that these algebras are all hyper-reflexive. [ABSTRACT FROM AUTHOR]

We establish a dilation-theoretic characterization of the Choquet order on the space of measures on a compact convex set using ideas from the theory of operator algebras. This yields an extension of Cartier's dilation theorem to the non-separable case, as well as a non-separable version of Šaškin's theorem from approximation theory. We show that a slight variant of this order characterizes the representations of a commutative C*-algebras that have the unique extension property relative to a set of generators. This reduces the commutative case of Arveson's hyperrigidity conjecture to the question of whether measures that are maximal with respect to the classical Choquet order are also maximal with respect to this new order. An example shows that these orders are not the same in general. [ABSTRACT FROM AUTHOR]