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(3 zwz − i zzw ) 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 ∞ ∞ M 2 � � f : n → M n n =1 n =1 where M n 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’ ⊙ s ℓ   ∞ m ∆ ℓ � � f ( X ) =  X − R f ( Y, . . . , Y ) , Y  ℓ =0 α =1

The nc universe The nc analogue of C d is ∞ M d def ( M n ) d . � = n =1 ⊕ defines a binary operation on M d : if x ∈ M n and y ∈ M m � � 0 x then x ⊕ y def = ∈ M n + m . 0 y If x = ( x 1 , . . . , x d ) and y = ( y 1 , . . . , y d ) are in M d then x ⊕ y def = ( x 1 ⊕ y 1 , . . . , x d ⊕ y d ) ∈ M d . Similarities: if s ∈ GL n ( C ) and x ∈ M d n then s − 1 xs def = ( s − 1 x 1 s, . . . , s − 1 x d s ) ∈ M d n .

Properties of the function f ( x 1 , x 2 ) = exp(3 x 1 x 2 x 1 − i x 1 x 1 x 2 ) The function f : M 2 → M 1 has three important properties. (1) f is graded : if x ∈ M 2 n then f ( x ) ∈ M n . (2) f preserves direct sums : f ( x ⊕ y ) = f ( x ) ⊕ f ( y ) for all x, y ∈ M 2 . (3) f preserves similarities : if s ∈ GL n ( C ) and x ∈ M 2 n then f ( s − 1 xs ) = s − 1 f ( x ) s.

nc functions An nc set is a subset of M d that is closed under ⊕ . An nc function is a function f defined on an nc set D ⊂ M d which is graded and preserves direct sums and similarities. n , s ∈ GL n ( C ) and s − 1 xs ∈ D then Thus, if x ∈ D ∩ M d f ( s − 1 xs ) = s − 1 f ( x ) s. Every free polynomial (that is, polynomial over C in d non- commuting indeterminates) defines an nc function on M d . An nc function f on D is analytic if D is open in the disjoint union topology on M d and f | D ∩ M d n is analytic for every n . Try to extend classical function theory to nc functions.

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

Free holomorphy A function f on a set D ⊂ M d is freely holomorphic if (1) D is a freely open set in M d (2) f is a freely locally nc function D → M 1 (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 U α of X to a set D α ⊂ M d . For charts α, β the transition map T αβ : α ( U α ∩ U β ) → β ( U α ∩ U β ) is T αβ = β ◦ α − 1 . A is a d -dimensional nc atlas for X if { U α : α ∈ A} covers X and, for all α, β ∈ A , (1) α ( U α ∩ U β ) is a union of nc sets and (2) the restriction of T αβ to any nc subset of α ( U α ∩ U β ) 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 M d and the transition maps T αβ are freely holomorphic for every α, β ∈ A . A map f : X → M 1 is a freely holomorphic function on the free manifold ( X, T , A ) if f ◦ α − 1 is a freely holomorphic function on D α for every α ∈ A .

The matricial square root function For x ∈ M n , √ x def = the set of y ∈ M n such that y 2 = 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 M 1 on which √· is nonempty everywhere. The free Riemann surface R of √·|I is a 1 - Theorem. 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 on 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 x 0 ∈ U ∩ V then f and g agree on some free neighbourhood of x 0 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, B δ ) over all basic freely open subsets B δ of I and all branches ( f, B δ ) of √· . Thus R ⊂ M 2 By the Lemma, the collection of sets { graph( f, B δ ) } where B δ is a basic freely open subset of I and ( f, B δ ) is a branch of √· , constitutes a base for a topology T on R . Let α fδ : graph( f, B δ ) → B δ be given by α fδ ( x, f ( x )) = x. Let A comprise all the charts α fδ . 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 α fδ are freely open in M 1 and the transition maps T αδ,βγ are freely holomorphic. α fδ is a bijective map from graph( f, B δ ) to the freely open nc subset B δ of I . Hence α fδ is an nc chart on R . The union of the domains graph( f, B δ ) is R , by definition. The transition function T gγ,fδ is the identity map on B δ ⊕ γ , a freely holomorphic nc map.

√· as a function on R Recall that R ⊂ M 2 . Let F be the restriction to R of the second co-ordinate projection on M 2 , that is, ( x 1 , x 2 ) �→ x 2 . 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 2 k square roots in the algebra generated by 1 and x . The number of sheets of the free Riemann surface of √· over a point x ∈ I is finite but unbounded. The end