A matrix convex set is a set of the form S = ∪ n≥1 Sn (where each Sn is a set of d-tuples of n×n matrices) that is invariant under UCP maps from Mn to M k 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 = ∪ n≥1 Sn, and T = ∪ n≥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 W max (B d ) constructed from the unit ball B d . 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) = ∪n{(T1, . . . , T d ) ∈ M d n : Ti ≤ 1}, is that {x ∈ R d : |xj | ≤ 1} ⊆ 1 d T1, i.e., that 1 d T1 contains the polar dual of the cube [−1, 1] d = C (d) 1 . 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.2010 Mathematics Subject Classification. 47A13, 47B32, 12Y05, 13P10. Key words and phrases. matrix convex set, free spectrahedra, matrix range, completely positive maps.

This paper addresses the isomorphism problem for the universal (nonself-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 radical 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 C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the weak-operator closures of these algebras as well.Comment: 46 pages. Final version, to appear in Advances in Mathematic

Abstract. We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions M V of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball B d .We find that M V is completely isometrically isomorphic to M W 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 strengthend 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 in B d with d < ∞, then an isomorphism between M V and M W 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.

We study algebras of bounded, noncommutative (nc) analytic functions on nc subvarieties of the nc unit ball. Given an nc variety V in the nc unit ball B d , we identify the algebra of bounded analytic functions on V -denoted H ∞ (V) -as the multiplier algebra Mult H V of a certain reproducing kernel Hilbert space H V consisting of nc functions on V. We find that every such algebra H ∞ (V) is completely isometrically isomorphic to the quotient H ∞ (B d )/J V of the algebra of bounded nc holomorphic functions on the ball by the ideal J V of bounded nc holomorphic functions which vanish on V. In order to demonstrate this isomorphism, we prove that the space H V is an nc complete Pick space (a fact recently proved -by other methods -by Ball, Marx and Vinnikov).We investigate the problem of when two algebras H ∞ (V) and H ∞ (W) are (completely) isometrically isomorphic. If the variety W is the image of V under an nc analytic automorphism of B d , then H ∞ (V) and H ∞ (W) are completely isometrically isomorphic. We prove that the converse holds in the case where the varieties are homogeneous; in general we can only show that if the algebras are completely isometrically isomorphic, then there must be nc holomorphic maps between the varieties (in the case d = ∞ we need to assume that the isomorphism is also weak- * continuous).We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of B d and related norm closed algebras; the results in the norm closed setting are somewhat simpler and work for the case d = ∞ without further assumptions.Along the way, we are led to consider some interesting problems on function theory in the nc unit ball. For example, we study various versions of the Nullstellensatz (that is, the problem of to what extent an ideal is determined by its zero set), and we obtain perfect Nullstellensatz in both the homogeneous as well as the commutative cases.Analogues of the Nevanlinna-Pick interpolation on the noncommutative ball first appeared in [21] and [12]. More general noncommutative versions of the classical interpolation and realization results appeared recently in the works of Agler and M c Carthy [5] and Ball, Marx and Vinnikov [14,15], who also introduced a generalization of reproducing kernel Hilbert spaces to the free setting.Our first goal in this work is to show that the full Fock space is a noncommutative reproducing kernel Hilbert space (nc RKHS), and its algebra of multipliers is, on the one hand, the algebra of bounded functions on the noncommutative ball (such that the multiplier norm and the supremum norm coincide), and, on the other hand, that this algebra coincides with the WOT closed algebra considered by Arias-Popescu and Davidson-Pitts. With this identification in hand, our second goal is to show that several results of [14] in the case of the noncommutative ball follow from established operator algebraic techniques and results, in particular, the complete Pick property of the noncommutative kernel of the full Fock space.W...

Abstract. To every convex body K ⊆ R d , one may associate a minimal matrix convex set W min (K), and a maximal matrix convex set W max (K), which have K as their ground level. The main question treated in this paper is: under what conditions on a given pair of convex bodies This constant is sharp, and it is new for all p = 2. Moreover, for some sets K we find a minimal set L for whichIn particular, we obtain that a convex body K satisfies W max (K) = W min (K) if and only if K is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. We discuss and exploit these connections as well. For example, our results show that every d-tuple of self-adjoint operators of norm less than or equal to 1, can be dilated to a commuting family of self-adjoints, each of norm at most √ d. We also introduce new explicit constructions of these (and other) dilations.