CHAPTER 1

Introduction

The goal of this work is to develop, in a systematic way and in a full natural

generality, the foundations of a theory of functions of free1 noncommuting vari-

ables. This theory offers a unified treatment for many free noncommutative objects

appearing in various branches of mathematics.

Analytic functions of d noncommuting variables originate in the pioneering

work of J. L. Taylor on noncommutative spectral theory [99, 100]. The underlying

idea is that a function of d noncommuting variables is a function on d-tuples of

square matrices of all sizes that respects simultaneous intertwinings (or equivalently

— as we will show — direct sums and simultaneous similarities). Taylor showed

that such functions admit a good differential (more precisely, difference-differential)

calculus, all the way to the noncommutative counterpart of the classical (Brook)

Taylor formula. Of course a d-tuple of matrices (say over C) is the same thing as

a matrix over

Cd,

so we can view a noncommutative function as defined on square

matrices of all sizes over a given vector space. This puts noncommutative function

theory in the framework of operator spaces [37, 78, 79]. Also, noncommutative

functions equipped with the difference-differential operator form an infinitesimal

bialgebra [90,

3].2

The theory has been pushed forward by Voiculescu [105, 106,

107], with an eye towards applications in free probability [102, 103, 104, 108]. We

mention also the work of Hadwin [46] and Hadwin–Kaonga–Mathes [47], of Popescu

[85, 86, 88, 89], of Helton–Klep–McCullough [55, 51, 52, 53], and of Muhly–

Solel [69, 71]. The (already nontrivial) case of functions of a single noncommutative

variable3 was considered by Schanuel [93] (see also Schanuel–Zame [94]) and by

Niemiec [76].

In a purely algebraic setting, polynomials and rational functions in d noncom-

muting indeterminates and their evaluations on d-tuples of matrices of an arbitrary

fixed size (over a commutative ring R) are central objects in the theory of polyno-

mial and rational identities; see, e.g., [91, 41]. A deep and detailed study of the

ring of noncommutative polynomials and the skew field of noncommutative rational

1We

consider only the case of free noncommuting variables, namely a free algebra or more

generally the tensor algebra of a module; we will therefore say simply “noncommutative” instead

of “free noncommutative”.

2More

precisely, to use our terminology, we have to consider noncommutative functions with

values in the noncommutative space over an algebra with a directional noncommutative difference-

differential operator as a comultiplication, and it is a topological version of the bialgebra concept

where the range of the comultiplication is a completed tensor product. See Section 2.3.4 for the

Leibnitz rule, and Section 3.4 for the coassociativity of the comultiplication. We will not pursue

the infinitesimal bialgrebra viewpoint explicitly.

3See

Remark 5.11.

1

http://dx.doi.org/10.1090/surv/199/01