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.

We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D ' abrowski, and Hajac. When a unital C * -algebra A admits a free action of Z/kZ, k ≥ 2, there is no equivariant map from A to the C * -algebraic join of A and the compact "quantum" group C(Z/kZ). This also resolves D ' abrowski's conjecture on unreduced suspensions of C * -algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions.

We study containment and uniqueness problems concerning matrix convex sets. First, to what extent is a matrix convex set determined by its first level? Our results in this direction quantify the disparity between two product operations, namely the product of the smallest matrix convex sets over K i ⊆ C d , and the smallest matrix convex set over the product of K i . Second, if a matrix convex set is given as the matrix range of an operator tuple T , when is T determined uniquely? We provide counterexamples to results in the literature, showing that a compact tuple meeting a minimality condition need not be determined uniquely, even if its matrix range is a particularly friendly set. Finally, our results may be used to improve dilation scales, such as the norm bound on the dilation of (non self-adjoint) contractions to commuting normal operators, both concretely and abstractly.

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.This notion of minimality was used in [8, Section 6], and in the finite-dimensional case it corresponds precisely to the notion of minimal pencil used earlier in [17] (and to the notion of σ-minimal pencil used in [23]). Using matricial polar duality [11], one can show that 2010 Mathematics Subject Classification. 47A20, 47A13, 46L07, 47L25. Key words and phrases. Matrix convex set; matrix range; matrix extreme point; operator system; structure of compact tuples.