Linear matrix inequalities (LMIs) I d + g j=1 Ajxj + g j=1 A * j x * j 0 play a role in many areas of applications. The set of solutions of an LMI is a spectrahedron. LMIs in (dimension-free) matrix variables model most problems in linear systems engineering, and their solution sets are called free spectrahedra. Free spectrahedra are exactly the free semialgebraic convex sets.This paper studies free analytic maps between free spectrahedra and, under certain (generically valid) irreducibility assumptions, classifies all those that are bianalytic. The foundation of such maps turns out to be a very small class of birational maps we call convexotonic. The convexotonic maps in g variables sit in correspondence with g-dimensional algebras. If two bounded free spectrahedra DA and DB meeting our irreducibility assumptions are free bianalytic with map denoted p, then p must (after possibly an affine linear transform) extend to a convexotonic map corresponding to a g-dimensional algebra spanned by (U − I)A1, . . . , (U − I)Ag for some unitary U . Furthermore, B and U A are unitarily equivalent.The article also establishes a Positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from DA to DB (not necessarily bianalytic). Another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron. These theorems are needed for classifying free bianalytic maps.

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Abstract. Subsets of the set of g-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, contained in the hull of a single point.

We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$ . In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.

The main result of this paper establishes the free analog of Grothendieck's theorem on bijective polynomial mappings of Cg. Namely, we show if p is a polynomial mapping in g freely non‐commuting variables sending g‐tuples of matrices (of the same size) to g‐tuples of matrices (of the same size) that is injective, then it has a free polynomial inverse. Other results include an algorithm that tests if a free polynomial mapping p has a polynomial inverse (equivalently is injective; equivalently is bijective). Further, a class of free algebraic functions, called hyporational, lying strictly between the free rational functions and the free algebraic functions are identified. They play a significant role in the proof of the main result.