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.

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 Viselter's 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 Ω, and let T+(X) and T+(Y ) be their tensor algebras. We show that every algebraic isomorphism from T+(X) onto T+(Y ) is automatically bounded. Furthermore, T+(X) and T+(Y ) are isometrically isomorphic if and only if X and Y are unitarily isomorphic up to a *-automorphism of ℓ ∞ (Ω). When Ω is finite, we prove that T+(X) and T+(Y ) are algebraically isomorphic if and only if there exists a similarity between X and Y up to a *-automorphism of ℓ ∞ (Ω). Moreover, we provide an explicit description of the Cuntz-Pimsner algebra O(X) in the case where Ω is finite and the stochastic matrix is essential.1 2 ADAM DOR-ON AND DANIEL MARKIEWICZ is easy to find two graphs with d vertices and irreducible adjacency matrix whose C*-algebras are not isomorphic (see [Rae05]). In contrast, in section 5 we show that if X is the Arveson-Stinespring subproduct system of a d × d irreducible stochastic matrix, then O(X) ∼ = C(T) ⊗ M d (C). More generally, we also provide an explicit description for the Cuntz-Pimsner algebra of a subproduct system associated to essential finite stochastic matrices.On the other hand, the non-self-adjoint tensor algebra T + (E) of a C*-correspondence E over A has often proven to be a strong invariant of the correspondence. Muhly and Solel [MS00] proved that if E and F are aperiodic C*-correspondences, then T + (E) is isometrically isomorphic to T + (F ) if and only if E and F are isometrically isomorphic as A-bimodules. Similarly, Katsoulis and Kribs [KK04] and Solel [Sol04] proved that if G and G ′ are countable directed graphs, then the tensor algebras T + (G) and T + (G ′ ) are isomorphic as algebras if and only if G and G ′ are isomorphic as directed graphs. See also Davidson and Katsoulis [DK11] for another important example of this phenomenon of increased acuity of the normed (non-self-adjoint) algebras as opposed to C*algebras, perhaps first recognized in Arveson [Arv67] and Arveson and Josephson [AJ69]. The tensor algebras of subproduct systems were first considered by Solel and Shalit [SS09] in the special case of M = C, and they analyzed the problem of graded isomorphism of their tensor algebras. The general isomorphism problem for such subproduct systems was resolved by Davidson, Ramsey and Shalit [DRS11]. They proved that if X, Y are subproduct systems of finitedimensional Hilbert space fibers, then T + (X) and T + (Y ) are isometrically isomorphic if and only if X and Y are (unitarily) isomorphic.On the other hand, the recent work of Gurevich [Gur12] provides a useful contrast. Although in this paper we focus on subproduct systems over N, it is possible to consider mo...

Let (G, P ) be an abelian, lattice ordered group and let X be a compactly aligned product system over P with coefficients in A. We show that the C*-envelope of the Nica tensor algebra N T + X coincides with both Sehnem's covariance algebra A ×X P and the co-universal C * -algebra N O r X for injective, gauge-compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of N O r X , thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C * -envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C *algebras of product systems. This generalizes recent results that were obtained by various authors in the case where (G, P ) = (Z, N).2010 Mathematics Subject Classification. Primary: 46L08, 46L55, 47B49, 47L40, 47L65, 46L05.

In this paper we study the C*-envelope of the (non-self-adjoint) tensor algebra associated via subproduct systems to a finite irreducible stochastic matrix P .Firstly, we identify the boundary representations of the tensor algebra inside the Toeplitz algebra, also known as its non-commutative Choquet boundary. As an application, we provide examples of C*-envelopes that are not *-isomorphic to either the Toeplitz algebra or the Cuntz-Pimsner algebra. This characterization required a new proof for the fact that the Cuntz-Pimsner algebra associated to P is isomorphic to C(T, M d (C)), filling a gap in a previous paper.We then proceed to classify the C*-envelopes of tensor algebras of stochastic matrices up to *isomorphism and stable isomorphism, in terms of the underlying matrices. This is accomplished by determining the K-theory of these C*-algebras and by combining this information with results due to Paschke and Salinas in extension theory. This classification is applied to provide a clearer picture of the various C*-envelopes that can land between the Toeplitz and the Cuntz-Pimsner algebras.

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