Aspects of Non-commutative Function Theory

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.


LMI's A linear matrix inequality (LMI) is an inequality of the form
(1. 2) The A i are given self-adjoint n-by-n matrices, and the object is to find x ∈ R m such that (1.2) is satisfied (or to show that it is never satisfied). LMI's are very important in control theory, and there are efficient algorithms for finding solutions. See for example the book [9]. Often, the stability of a system is equivalent to whether a certain matrix valued function F (x) is positive semi-definite; but the function F may be non-linear. A big question is when the inequality can be reduced to an LMI. This has been studied, for example, in [15,16,18,19]. Let us consider a simple example, the Riccati matrix inequality: AX + XA * − XBB * X + C * C > 0. (1.3) All the matrices in (1.3) are real, and A, B, C are known. The self-adjoint matrix X is unknown. The quadratic inequality (1.3) can be transformed into the following LMI: AX + XA * + C * C XB B * X I > 0. (1.4) Suppose each matrix A, B, C, X is 3-by-3. Then, in terms of the matrix entries of X, (1.4) is an LMI in 6 variables. If one uses Sylvester's criterion to determine positive definiteness, one gets that (1.4) is equivalent to the positivity of 6 polynomials, of degrees 1 through 6, in these 6 variables. This could then become a question in real algebraic geometry. However, this approach has two obvious flaws: it loses the matrix structure, and the complexity increases rapidly with the dimension. Can one study inequalities like (1.4) in a dimension independent way?

Non-commutative sums of squares
Hilbert's seventeenth problem asked whether a non-negative polynomial in real variables could be written as a sum of squares of rational functions (Hilbert knew that a sum of squares of polynomials did not in general suffice). Although the answer was proved to be yes by E. Artin in 1927, such problems are still an active area of research in real algebraic geometry today, and operator theory and functional analysis have played a rôle in this field [30,31].
Let M n denote the n-by-n matrices. J.W. Helton asked whether a noncommutative polynomial p in the 2d variables x 1 , . . . , x d , x * 1 , . . . , x * d that is formally self-adjoint, and has the property that as a (finite) sum of squares, where each q j is a non-commutative polynomial in d variables. He and S. McCullough proved [12,20] that the answer is yes.
This illustrates an important theme: if something is true at all matrix levels (a strong assumption), it is true for obvious algebraic reasons (a strong conclusion), and the proof is often easier than in the scalar case.

Implicit Function Theorem
Consider the matrix equation If X is similar to a diagonal matrix, it is obvious that one can find solutions Y to (1.5) that commute with it. What is perhaps surprising is that generically, this is all that happens: Proposition 1.6. For a generic choice of X ∈ M n , the only Y 's that satisfy (1.5) commute with X.
To see why this is true (and what we mean by generic) let p(X, Y ) = X 3 + 2X 2 Y + 3Y X + 4X + 5Y + 6, and consider the partial derivative of p w.r.t. Y in the direction H. This is An implicit function theorem would say that we can write Y locally as a function of X (and therefore as a matrix that commutes with X) provided ∂ ∂Y p(X, Y ) is full rank. Ontologically, the partial derivative is a linear map from M n to M n . It is therefore full rank if and only if it has no kernel. We can analyze this case using the following theorem of Sylvester from 1884 [32]: Condition (1.8) in turn is generically true, so for such X Proposition 1.6 will follow from a non-commutative implicit function, such as Theorem 3.7 below.

Functional Calculus
If T is a single operator on a Banach space E, the functional calculus is an extension of the evaluation map to a larger algebra than the polynomials. The Riesz-Dunford functional calculus gives the unique extension to the algebra of all functions that are holomorphic on a neighborhood of σ(T ). If now T = (T 1 , . . . , T d ) is a commuting d-tuple of operators, then the Taylor functional calculus extends the evaluation map from polynomials (in d commuting variables) to functions holomorphic on a neighborhood of the Taylor spectrum of T [33,34].
What about the case that T is a d-tuple of non-commuting operators on X? Two distinct questions arise.
(i) What should play the rôle of holomorphic functions in the non-commutative setting?
(ii) How can they be applied to elements of L(E) d ?
The answers to both questions have their roots in work of Taylor shortly after he completed his work on commuting operators [35,36]. The answer to the first question is nc-functions, which we shall discuss in Section 2, and is the subject of the recent book [21] by D. Kaliuzhnyi-Verbovetskyi and V. Vinnikov. The second question is discussed in Section 4.

Other motivations
There has been an upwelling of interest in non-commutative function theory recently. In addition to the motivations above, let us mention the work of Voiculescu [37], in the context of free probability; Popescu [26][27][28][29], in the context of extending classical function theory to d-tuples of bounded operators; Ball, Groenewald and Malakorn [8], in the context of extending realization formulas from functions of commuting operators to functions of non-commuting operators; Alpay and Kalyuzhnyi-Verbovetzkii [5] in the context of realization formulas for rational functions that are J-unitary on the boundary of the domain; and Helton, Klep and McCullough [13,14] and Helton and McCullough [18] in the context of developing a descriptive theory of the domains on which LMI and semi-definite programming apply; Muhly and Solel [23], in the context of tensorial function theory; Cimpric, Helton, McCullough and Nelson [10] in the context of non-commutative real Nullstellensätze; the second author and Timoney [22] and of Helton, Klep, McCullough and Slinglend, in [17] on non commutative automorphisms; and the work of Pascoe and Tully-Doyle [25] on non-commutative operator monotonicity.
The book [21] by D. Kaliuzhnyi-Verbovetskyi and V. Vinnikov is a much more comprehensive treatment of the subject than we can give here, and also contains extensive historical references.

NC-functions
We want to evaluate functions on d-tuples of matrices, where all the matrices in a given d-tuple are the same dimension, but we want to allow this dimension to vary. So our universe becomes We shall call a function f defined on a subset of is a graded function that respects direct sums and joint similiarities, i.e.
where the equations are only required to hold when the arguments on both sides are in Ω.
Notice that every non-commutative polynomial is an nc-function on M [d] .
To get utility from the definition, one needs Ω to have some structure.
n and u is an n-by-n unitary, then u * xu ∈ Ω. The algebraic properties of being nc on an nc-set have analytic consequences. For example, J.W. Helton, I. Klep and S. McCullough in [14] proved that continuity implies analyticity (continuity can actually be weakened to local boundedness). x Since f is nc, Proof of Theorem 2.3. First, we show that local boundedness implies continuity. Fix x ∈ Ω ∩ M d n and let ε > 0. Choose r > 0 so that the ball If r 1 is chosen with 0 < r 1 < r then as B(x ⊕ x, r 1 ) − is a compact subset of Ω ∩ M d 2n and f is assumed locally bounded, there exists a constant C such that Choose δ sufficiently small so that δ < min{r 1 ε/2C, r 1 /2} and B(x, δ) ⊆ Ω. That f is continuous at x follows from the following claim.
To see that f is holomorphic, fix x ∈ Ω ∩ M d n . If h ∈ M d n is selected sufficiently small, then for all sufficiently small λ ∈ C. But Hence, Lemma 2.4 implies that As the left hand side of (2.8) is continuous at λ = 0, it follows that the 1-2 entry of the right hand side of (2.8) must converge. As h is arbitrary after scaling, this implies that f is holomorphic. 2

τ -holomorphic functions
The theory of holomorphic functions on infinite dimensional spaces depends heavily on the topologies chosen -see e.g. [11]. For nc-functions too, the topology matters.  By Theorem 2.3, every τ -holomorphic function on Ω is, at each level n, an analytic function from Ω ∩ M d n to M n .

Example 3.3
The fine topology is the admissible topology generated by all nc-sets. Since this is the largest admissible topology, for any admissible topology τ , any τ -holomorphic function is automatically fine holomorphic. The class of nc functions considered in [14,24] is the fine holomorphic functions.
J. Pascoe proved the following inverse function theorem in [24]. The equivalence of (i) and (iii) is due to Helton, Klep and McCullough [14]. (i) Φ is injective on Ω.
(iii) The function Φ −1 exists and is a fine holomorphic map.
Example 3.5 Let R + = {r ∈ R | r > 0}. For n ∈ N, a ∈ M d n , and r ∈ R + , we let D n (a, r) ⊆ M d n be the matrix polydisc defined by If a ∈ M d n , r ∈ R + , we define D(a, r) ⊆ M d by where a (k) denotes the direct sum of k copies of a. Finally, if a ∈ M d , r ∈ R + , we define F (a, r) ⊆ M d by where U m denotes the set of m × m unitary matrices. It can be shown [4] that the sets F (a, r) are nc sets that form the basis for a topology. We call this topology the fat topology.
There is an implicit function theorem for both the fine and fat topologies. The hypotheses require the derivative to be full rank on a neighborhood of the point; the advantage of the fat topology is that if the derivative is full rank at one point, it is automatically full rank on a neighborhood [4, Thm.

5.5].
Theorem 3.7. Let Ω an nc domain. Let f = (f 1 , . . . , f k ) t be a vector of k fine holomorphic function on Ω, for some 1 ≤ k ≤ d − 1 (equivalently, a L(C, C k ) valued fine holomorphic function). Suppose Let W be the projection onto the first d − k coordinates of Z f ∩ Ω. Then there is an L(C, C k )-valued fine holomorphic function g on W such that Moreover, if f is fat holomorphic, then g can also be taken to be fat holomorphic.
where δ is a J-by-J matrix with entries in P d . We define the free topology to be the topology on M [d] which has as a basis all the sets G δ , as J ranges over the positive integers, and the entries of δ range over all polynomials in P d . (Notice that G δ 1 ∩G δ 2 = G δ 1 ⊕δ 2 , so these sets do form the basis of a topology). The free topology is a natural topology when considering semi-algebraic sets. Example 3.9 Another admissible topology is the Zariski free topology, which is generated by sets of the form There is no Goldilocks topology. The free topology has good polynomial approximation properties -not only is every free holomorphic function pointwise approximable by free polynomials, there is an Oka-Weil theorem which says that on sets of the form G δ , bounded free holomorphic functions are uniformly approximable on compact sets by free polynomials [2]. In contrast, neither the fine nor fat topology admit even pointwise polynomial approximation. Indeed, one cannot have an admissible topology for which there is both an implicit function theorem and pointwise approximation by polynomials, as the following example shows (a formal statement is in [4]).  It is immediate from (3.13) that ∂ ∂Z p(a) : M 2 → M 2 is onto, and so has a right inverse. By [4,Thm 5.5], there is a fat domain Ω a such that ∂ ∂Z p(λ) is non-singular for all λ ∈ Ω. So by the Implicit Function Theorem 3.7, if W is the projection onto the first two coordinates of Ω, then W is a fat domain containing x 0 and there is a fat holomorphic function g defined on W such that g(x 0 ) = z 0 . Since z 0 is not in the algebra generated by x 0 , we cannot approximate g pointwise by free polynomials.

Free holomorphic functions
We shall let H ∞ (G δ ) denote the bounded free holomorphic functions on the polynomial polyhedron G δ . Just as in the commutative case [6,7], there is a realization formula for these functions. : An immediate corollary is that if one truncates the Neumann series of −1 one gets a sequence of free polynomials that converges to f uniformly on G tδ for every t > 1.
The space K that appears in (4.2) is finite dimensional, but the formula would make sense even it were not. Let us fix an infinite dimensional separable Hilbert space H. Let us define G δ by If one extends formula (4.2) to G δ , what sort of functions does one get?
Let Ω ⊆ L(H) d , and let F : Ω → L(H). We say that F is intertwining preserving (IP) if: (i) Whenever x, y ∈ Ω and there exists some bounded linear operator T ∈ L(H) such that T x = yT , then T F (x) = F (y)T .
(ii) Whenever (x n ) is a bounded sequence in Ω, and there exists some invertible bounded linear operator s : H → ⊕H such that F (x 2 ) · · · · · · · · · . . .  Since multiplication is sequentially strong operator continuous, it follows that every free polynomial is SSOC, and this property is also inherited by limits on sets that are closed w.r.t. direct sums. The following theorem from [3] characterizes what functions arise in (4.2): Theorem 4.5. Assume that G δ is connected and contains 0. Let F : G δ → L 1 (H) be sequentially strong operator continuous. Then the following are equivalent: (i) The function F is intertwining preserving.
(ii) For each t > 1, the function F is uniformly approximable by free polynomials on G tδ .  It could be the case that a free holomorphic function on G δ could have more than one extension to G δ . For example, if δ(x 1 , x 2 ) = x 1 x 2 − x 2 x 1 − 1, then G δ is empty, but G δ is not. The following theorem is from [3].
Theorem 4.7. Assume that G δ is connected and contains 0. Then every bounded free holomorphic function on G δ has a unique extension to an IP SSOC function on G δ .
What about applying free holomorphic functions to d-tuples of operators on a Banach space, L(X)? This would allow the development of a noncommutative functional calculus. This is considered in [1]. The basic idea is to take a reasonable cross-norm h defined on X ⊗K for every Hilbert space K. Let (T ) = (T ij ) be an I-by-J matrix of elements of L(X). Let E ij : C J → C I be the matrix with 1 in the (i, j) entry and zero elsewhere. Then we get a norm on (T ) by where the ⊗ without a subscript means the Hilbert space tensor product. We define (T ) • := inf h (T ) h .
Then provided δ(T ) • < 1, we can evaluate f (T ), when f is in H ∞ (G δ ), by a formula like (4.2) (or (4.6)). Since tensor products need not be associative, we do all the Hilbert space tensor products first, and then tensor with the L(X) operator. Here is one theorem from [1].
Then there exists a unique completely bounded homomorphism π : H ∞ (G δ ) → L(X) that extends the evaluation map on free polynomials p → p(T ).