Progress in noncommutative function theory

Abstract

In this expository paper we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of representations. We will show that these spaces of representations can be parameterized as unit balls of certain $W^{\ast }$-correspondences and the functions can be viewed as Schur class operator functions on these balls. We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory.

References in corpus (3)

• Free Semigroupoid Algebras
• Proper Analytic Free Maps
• Free holomorphic functions on the unit ball of B(H)^n

Cited by in corpus (10)

• Free function theory through matrix invariants
• Extreme points of matrix convex sets, free spectrahedra and dilation theory
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• Free Convex Algebraic Geometry
• Noncommutative polynomials nonnegative on a variety intersect a convex set
• Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation
• Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras
• Noncommutative hyperbolic metrics
• Integrability of Free Noncommutative Functions
• Fixed point theorems for noncommutative functions