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
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,
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
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”.
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.
Remark 5.11.
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