No abstract

Given a von Neumann algebra M and a W * -correspondence E over M , we construct an algebra H ∞ (E) that we call the Hardy algebra of E. When M = C = E, then H ∞ (E) is the classical Hardy space H ∞ (T) of bounded analytic functions on the unit disc. We show that given any faithful normal representation σ of M on a Hilbert space H there is a natural correspondence E σ over the commutant σ(M ) ′ , called the σ-dual of E, and that H ∞ (E) can be realized in terms of (B(H)-valued) functions on the open unit ball D((E σ ) * ) in the space of adjoints of elements in E σ . We prove analogues of the Nevanlinna-Pick theorem in this setting and discover other aspects of the value Hardy algebras. That is, if we take a (faithful) normal representation σ of M on a Hilbert space H and form the dual E σ , getting a W * -correspondence over σ(M) ′ and if we then form the dual of E σ with respect to the identity representation of σ(M) ′ , we get a W * -correspondence E σ,ι over σ(M) ′′ = σ(M) that is naturally isomorphic to E. (See Theorem 3.6.) Further, V × ρ andṼ ×ρ are ultraweak homeomorphisms and (As we will see in Theorem 2.13, given a faithful normal representation σ of M on a Hilbert space H, we may evaluate elements in H ∞

We study the structure of quantum Markov Processes from the point of view of product systems and their representations.

Abstract. Our objective in this sequel to [18] is to develop extensions, to representations of tensor algebras over C * -correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling's theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.

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.