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...

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety X ⊂ P d of an arbitrary codimension ℓ with respect to a real ℓ − 1-dimensional linear subspace V ⊂ P d and study its basic properties. We also consider a special kind of determinantal representations that we call Livsic-type and a nice subclass of these that we call very reasonable. Much like in the case of hypersurfaces (ℓ = 1), the existence of a definite Hermitian very reasonable Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a very reasonable Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in P d hyperbolic with respect to some real d − 2-dimensional linear subspace admits a definite Hermitian, or even real symmetric, very reasonable Livsic-type determinantal representation.

In this paper, we define and study real fibered morphisms. Such morphisms arise in the study of real hyperbolic hypersurfaces in P d and other hyperbolic varieties. We show that real fibered morphisms are intimately connected to Ulrich sheaves admitting positive definite symmetric bilinear forms.

We characterize the non-commutative Aleksandrov-Clark measures and the minimal realization formulas of contractive and, in particular, isometric non-commutative rational multipliers of the Fock space. Here, the full Fock space over C is defined as the Hilbert space of square-summable power series in several non-commuting formal variables, and we interpret this space as the noncommutative and multi-variable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov-Clark measure theory for non-commutative and contractive rational multipliers. Non-commutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz-Toeplitz algebra, the unital * −algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz-Toeplitz and Cuntz algebras and the emerging field of non-commutative rational functions.

We introduce a notion of noncommutative Choquet simplex, or briefly an nc simplex, that generalizes the classical notion of a simplex. While every simplex is an nc simplex, there are many more nc simplices. They arise naturally from C*-algebras and in noncommutative dynamics. We characterize nc simplices in terms of their geometry and in terms of structural properties of their corresponding operator systems.There is a natural definition of nc Bauer simplex that generalizes the classical definition of a Bauer simplex. We show that a compact nc convex set is an nc Bauer simplex if and only if it is affinely homeomorphic to the nc state space of a unital C*-algebra, generalizing a classical result of Bauer for unital commutative C*algebras.We obtain several applications to noncommutative dynamics. We show that the set of nc states of a C*-algebra that are invariant with respect to the action of a discrete group is an nc simplex. From this, we obtain a noncommutative ergodic decomposition theorem with uniqueness.Finally, we establish a new characterization of discrete groups with Kazhdan's property (T) that extends a result of Glasner and Weiss. Specifically, we show that a discrete group has property (T) if and only if for every action of the group on a unital C*-algebra, the set of invariant states is affinely homeomorphic to the state space of a unital C*-algebra.