Noncommutative analysis Nicholas Young Leeds and Newcastle - - PowerPoint PPT Presentation

Noncommutative analysis

Nicholas Young Leeds and Newcastle Universities Joint work with Jim Agler, UCSD Durham, April 2018

What is an analytic function of noncommuting variables?

The function f(z, w) = exp(3zwz − izzw) looks like an analytic function of noncommuting variables z and w. How should we interpret this statement?

• J. L. Taylor, Functions of several non-commuting variables,
• Bull. AMS 79 (1973)

interpreted f as a map f :

∞

• n=1

M2

n → ∞

• n=1

Mn where Mn denotes the algebra of n × n matrices over C.

Motives for noncommutative analysis

Joseph Taylor: construct a ‘spectral theory’ for noncom- muting tuples of operators. Dan Voiculescu: free probability William Helton: optimization problems in engineering Gelu Popescu: generalization of operator theory from com- muting to noncommuting tuples

A research monograph

‘Foundations of free noncommutative function theory’ by Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, AMS, 2014, gives the recent theory and many applications. The authors make extensive use of the ‘Taylor Taylor series’ f(X) =

∞

• ℓ=0

 X −

m

• α=1

Y

 

⊙sℓ

∆ℓ

Rf(Y, . . . , Y ),

The nc universe

The nc analogue of Cd is Md def =

∞

• n=1

(Mn)d. ⊕ defines a binary operation on Md: if x ∈ Mn and y ∈ Mm then x ⊕ y def =

• x

y

• ∈ Mn+m.

If x = (x1, . . . , xd) and y = (y1, . . . , yd) are in Md then x ⊕ y def = (x1 ⊕ y1, . . . , xd ⊕ yd) ∈ Md. Similarities: if s ∈ GLn(C) and x ∈ Md

n then

s−1xs def = (s−1x1s, . . . , s−1xds) ∈ Md

n.

Properties of the function f(x1, x2) = exp(3x1x2x1 − ix1x1x2)

The function f : M2 → M1 has three important properties. (1) f is graded: if x ∈ M2

n then f(x) ∈ Mn.

(2) f preserves direct sums: f(x ⊕ y) = f(x) ⊕ f(y) for all x, y ∈ M2. (3) f preserves similarities: if s ∈ GLn(C) and x ∈ M2

n then

f(s−1xs) = s−1f(x)s.

nc functions

An nc set is a subset of Md that is closed under ⊕. An nc function is a function f defined on an nc set D ⊂ Md which is graded and preserves direct sums and similarities. Thus, if x ∈ D ∩ Md

n, s ∈ GLn(C) and s−1xs ∈ D then

f(s−1xs) = s−1f(x)s. Every free polynomial (that is, polynomial over C in d non- commuting indeterminates) defines an nc function on Md. An nc function f on D is analytic if D is open in the disjoint union topology on Md and f|D ∩ Md

n is analytic for every n.

Try to extend classical function theory to nc functions.

The free topology on Md

For any I × J matrix δ = [δij] of free polynomials in d non- commuting variables define Bδ = {x ∈ Md : δ(x) < 1}. The free topology on Md is the topology for which a base consists of the sets Bδ. The free topology is not Hausdorff. It does not distinguish between x and x ⊕ x. Md is connected in the free topology.

Free holomorphy

A function f on a set D ⊂ Md is freely holomorphic if (1) D is a freely open set in Md (2) f is a freely locally nc function D → M1 (3) f is freely locally bounded on D. Surprising theorem A freely holomorphic function is analytic.

nc manifolds

Let X be a set. A d-dimensional nc chart on X is a bijective map α from a subset Uα of X to a set Dα ⊂ Md. For charts α, β the transition map Tαβ : α(Uα ∩Uβ) → β(Uα ∩ Uβ) is Tαβ = β ◦ α−1. A is a d-dimensional nc atlas for X if {Uα : α ∈ A} covers X and, for all α, β ∈ A, (1) α(Uα ∩ Uβ) is a union of nc sets and (2) the restriction of Tαβ to any nc subset of α(Uα ∩ Uβ) is an nc map. (X, A) is a d-dimensional nc manifold if A is a d-dimensional nc atlas for X.

Free manifolds

Let (X, A) be a d-dimensional nc manifold and let T be a topology on X. (X, T , A) is a d-dimensional free manifold if the range of every chart α ∈ A is freely open in Md and the transition maps Tαβ are freely holomorphic for every α, β ∈ A. A map f : X → M1 is a freely holomorphic function on the free manifold (X, T , A) if f ◦ α−1 is a freely holomorphic function on Dα for every α ∈ A.

The matricial square root function

For x ∈ Mn, √x def = the set of y ∈ Mn such that y2 = x and y is in the algebra generated by 1 and x. We construct a ‘free Riemann surface’ R for √·, analogous to the classical Riemann surface for √z. Let I denote the set of nonsingular matrices. I is the largest freely open set in M1 on which √· is nonempty everywhere. Theorem. The free Riemann surface R of √·|I is a 1- dimensional free manifold that lies over I. The multivalued function √· determines a single-valued freely holomorphic function on R.

Function elements

A free function element over I is a pair (f, U) where U is a freely open subset of I and f is a freely holomorphic function

• n U. We say that (f, U) is a branch of √· if f(x)2 = x for

all x ∈ U.

• Lemma. If (f, U) and (g, V ) are branches of √· over I which

agree at some point x0 ∈ U ∩ V then f and g agree on some free neighbourhood of x0 in U ∩ V . The free Riemann surface of √· will be obtained by the gluing together of the graphs of function elements, as is done classically in standard texts (e.g. Ahlfors).

The definition of R

Let R =

• graph(f, Bδ)
• ver all basic freely open subsets Bδ of I and all branches

(f, Bδ) of √· . Thus R ⊂ M2 By the Lemma, the collection of sets {graph(f, Bδ)} where Bδ is a basic freely open subset of I and (f, Bδ) is a branch

• f √· , constitutes a base for a topology T on R.

Let αfδ : graph(f, Bδ) → Bδ be given by αfδ(x, f(x)) = x. Let A comprise all the charts αfδ.

• Theorem. (R, T , A) is a free manifold.

Proof

Certainly T is a topology on R. We must show that A is an nc atlas on R, the ranges of the charts αfδ are freely open in M1 and the transition maps Tαδ,βγ are freely holomorphic. αfδ is a bijective map from graph(f, Bδ) to the freely open nc subset Bδ of I. Hence αfδ is an nc chart on R. The union of the domains graph(f, Bδ) is R, by definition. The transition function Tgγ,fδ is the identity map on Bδ⊕γ, a freely holomorphic nc map.

√· as a function on R

Recall that R ⊂ M2. Let F be the restriction to R of the second co-ordinate projection on M2, that is, (x1, x2) → x2. Theorem F is a freely holomorphic function on the free manifold R. For any point (x, y) ∈ R, F(x, y) ∈ √x. For every branch (f, U) of √·, the restriction of F to graph(f, U) agrees with f when x is identified with (x, f(x)). That F is freely holomorphic ≡ every branch of √· is freely holomorphic.

How many sheets?

The classical Riemann surface of √z has two sheets. A nonsingular matrix x with k distinct eigenvalues has ex- actly 2k square roots in the algebra generated by 1 and x. The number of sheets of the free Riemann surface of √·

• ver a point x ∈ I is finite but unbounded.

The end