Symmetric functions of two noncommuting variables Nicholas Young Leeds and Newcastle Universities Joint work with Jim Agler, UCSD Newcastle, November 2013
Abstract We prove a noncommutative analogue of the fact that every symmetric analytic function of ( z, w ) in the bidisc D 2 can be expressed as an analytic function of the variables z + w and zw . We construct an analytic nc-map π from the biball to an infinite-dimensional nc-domain Ω with the property that, for every bounded symmetric analytic function ϕ of two non- commuting variables on the biball, there exists a bounded analytic nc-function Φ on Ω such that ϕ = Φ ◦ π . We also establish a realization formula for Φ, and hence for ϕ , in terms of operators on Hilbert space.
Symmetric polynomials Every symmetric polynomial in two commuting variables z and w can be written as a polynomial in the variables z + w and zw . What if z and w do not commute? The polynomial zwz + wzw in non-commuting variables z, w cannot be written as p ( z + w, zw + wz ) for any nc-polynomial p .
Proof Suppose that zwz + wzw = f ( z + w, zw + wz ) j f j x j . Choose for some polynomial f and that f ( x, 0) = � � � � � 1 0 0 0 z = w = ; then , 0 − 1 1 0 � � � � 0 0 1 0 zw + wz = 0 , zwz + wzw = z + w = , . − 1 0 1 − 1 Note that ( z + w ) 2 = 1 2 . Hence � j � � � � � 0 0 1 0 1 0 � = = A 1 2 + B f j − 1 0 1 − 1 1 − 1 j for some A, B ∈ C , which is impossible.
Theorem (Margarete Wolf, 1936) There is no finite basis for the ring of symmetric polynomials in d nc indeterminates over C when d > 1. Here ‘nc’ means noncommuting or noncommutative . Polynomial nc-functions Let M n denote the space of complex n × n complex matrices. If f is a polynomial in nc indeterminates x 1 , . . . , x d then f determines a map ∞ ∞ f ♯ : ( M n ) d → � � M n n =1 n =1 by substitution, as in the previous example. e: f = g if and only if f ♯ = g ♯ . Poincar´
Properties of polynomial nc functions Regard the space ∞ M d def ( M n ) d � = n =1 as the nc analogue of C d . Endow M d with the discrete union topology. If f is a polynomial in d nc indeterminates then f ♯ satisfies 1) f ♯ maps ( M n ) d to M n for each n ≥ 1; 2) f ♯ ( x ⊕ y ) = f ♯ ( x ) ⊕ f ♯ ( y ) for all x, y ∈ M d ; 3) f ♯ ( s − 1 xs ) = s − 1 f ♯ ( x ) s for all n ≥ 1 and all s ∈ M n , , x ∈ ( M n ) d with s invertible. Here s − 1 ( x 1 , . . . , x d ) s means ( s − 1 x 1 s, . . . , s − 1 x d s ).
Analytic nc functions J. L. Taylor, 1974. Let Ω ⊂ M d be an nc domain. An analytic nc function on Ω is a map f : Ω → M 1 such that 1) f maps Ω ∩ ( M n ) d analytically to M n (“ f is graded”); 2) f ( x ⊕ y ) = f ( x ) ⊕ f ( y ) for all x, y ∈ Ω (“ f respects direct sums”); 3) f ( s − 1 xs ) = s − 1 f ( x ) s for all x ∈ Ω and all invertible s such that s − 1 xs ∈ Ω (“ f respects similarities”). An nc domain is an open subset of M d that is closed under direct sums and unitary similarity.
Symmetric analytic functions on the bidisc Let π : C 2 → C 2 be given by π ( z, w ) = ( z + w, zw ) . If ϕ : D 2 → C is analytic and symmetric in z and w then there exists an analytic function Φ : π ( D 2 ) → C such that ϕ = Φ ◦ π : π D 2 π ( D 2 ) − → ϕ ց ւ Φ C Are there analogous statements for symmetric functions of non-commuting variables? What is the analogue of π ( D 2 )?
The biball B 2 is the non-commutative analogue of the bidisc: ∞ B 2 def B n × B n ⊂ M 2 � = n =1 where B n denotes the open unit ball of the space M n of n × n complex matrices. It is an nc-domain: n =1 M 2 1) it is open in � ∞ n ; 2) if x = ( x 1 , x 2 ) ∈ B 2 and y = ( y 1 , y 2 ) ∈ B 2 then x ⊕ y def = ( x 1 ⊕ y 1 , x 2 ⊕ y 2 ) ∈ B 2 ; 3) if x ∈ B 2 ∩ M 2 n and u is an n × n unitary then ( u ∗ x 1 u, u ∗ x 2 u ) ∈ B 2 .
Symmetric analytic nc-functions on B 2 There is an nc-domain Ω in the space ∞ M ∞ def M ∞ � = n n =1 and an analytic nc-map π : B 2 → Ω (given by a simple ratio- nal expression) such that every bounded symmetric analytic nc-function ϕ on B 2 factors through π , and conversely. π B 2 Ω − → ϕ ց ւ Φ n M n � Φ can be expressed by means of a non-commutative ver- sion of the familiar linear fractional realization formula for functions in the Schur class.
Theorem There exists an nc-domain Ω in M ∞ such that π : B 2 → M ∞ which maps x �→ ( u, v 2 , vuv, vu 2 v, . . . ) , where u = x 1 + x 2 v = x 1 − x 2 , , 2 2 has the following three properties. 1) π is an analytic nc-map from B 2 to Ω; 2) for every bounded symmetric analytic nc-function ϕ on the biball there exists an analytic nc-function Φ on Ω such that ϕ = Φ ◦ π ; 3) for every g ∈ Ω and every contraction T the operator g ( T ) exists.
Ω and π Ω ⊂ M ∞ is the ‘open unit ball of the nc disc algebra’. A sequence ( g 0 , g 1 , g 2 , . . . ) ∈ M ∞ n belongs to Ω if and only if g ( z ) = g 0 + g 1 z + g 2 z 2 + . . . is the Taylor series of an n × n -matrix function g that is analytic in the open unit disc D and continous on the closure of D , and moreover satisfies sup z ∈ D � g ( z ) � < 1. Since π ( x ) = ( u, v 2 , vuv, vu 2 v, . . . ), π ( x )( z ) = u + v 2 z + vuvz 2 + vu 2 vz 3 + . . . = u + vz (1 + uz + u 2 z 2 + . . . ) v = u + vz (1 − uz ) − 1 v � − 1 x 1 − x 2 = x 1 + x 2 1 − x 1 + x 2 + x 1 − x 2 � z z . 2 2 2 2
The functional calculus Let g = ( g 0 , g 1 , g 2 , . . . ) ∈ M ∞ n . Identify g with the formal power series � ∞ j =0 g j z j . Then, for any matrix or bounded linear operator T we may write down the series � ∞ j =0 g j ⊗ T j . If T acts on a linear space H then the sum g ( T ) of the series, if the series converges in an appropriate topology, is an operator on C n ⊗ H . The map g �→ g ( T ) is called the functional calculus ; it is defined (for a given operator T ) on a subset of M ∞ . 3) says: π ( x )( T ) is defined [converges] for any x ∈ B 2 and any operator T such that � T � ≤ 1.
A realization formula For any symmetric analytic nc-function ϕ on the biball there exist a unitary operator U on ℓ 2 and a contractive operator � � a B : C ⊕ ℓ 2 → C ⊕ ℓ 2 C D such that the function Φ on Ω defined, for n ≥ 1 and g ∈ Ω ∩ M ∞ n , by Φ( g ) = a 1 n + ( 1 n ⊗ B ) g ( U ) ( 1 − ( 1 n ⊗ D ) g ( U )) − 1 ( 1 n ⊗ C ) is an nc-function satisfying ϕ = Φ ◦ π .
Another formulation Identify a sequence ( g j ) ∈ M ∞ with the formal power series j ≥ 0 g j z j . Then � � − 1 x 1 − x 2 π ( x )( z ) = x 1 + x 2 + x 1 − x 2 1 − x 1 + x 2 � z z . 2 2 2 2 For x ∈ B 2 ∩ M 2 n and a unitary operator U on ℓ 2 , π ( x )( U ) = u ⊗ 1 + v 2 ⊗ U + vuv ⊗ U 2 + . . . = x 1 + x 2 x 1 − x 2 � � ⊗ 1 ℓ 2 + ⊗ U × 2 2 � − 1 � 1 C n ⊗ ℓ 2 − x 1 + x 2 x 1 − x 2 � � ⊗ U ⊗ 1 ℓ 2 , 2 2 an operator on C n ⊗ ℓ 2 .
Realization again Let ϕ be a symmetric nc-function on B 2 bounded by 1 in norm. There exist a unitary U on ℓ 2 and a contraction � � a B : C ⊕ ℓ 2 → C ⊕ ℓ 2 C D such that, for x ∈ B 2 ∩ M 2 n , ϕ ( x ) = a 1 n + ( 1 n ⊗ B ) π ( x )( U ) × ( 1 − ( 1 n ⊗ D ) π ( x )( U )) − 1 ( 1 n ⊗ C ) where π ( x )( U ) = x 1 + x 2 x 1 − x 2 � � ⊗ 1 ℓ 2 + ⊗ U × 2 2 � − 1 � 1 C n ⊗ ℓ 2 − x 1 + x 2 x 1 − x 2 � � ⊗ U ⊗ 1 ℓ 2 . 2 2
A final question about symmetric polynomials The polynomials u, v 2 , vuv, vu 2 v, . . . constitute a basis for the algebra of symmetric polynomials in x 1 , x 2 , where 2 ( x 1 + x 2 ) , 2 ( x 1 − x 2 ) . u = 1 v = 1 Observe that vu 2 v = vuv ( v 2 ) − 1 vuv, vu 3 v = vuv ( v 2 ) − 1 vuv ( v 2 ) − 1 vuv, . . . Hence the algebra of rational symmetric functions in two variables has a basis consisting of the three functions u, v 2 , vuv . Is there an analogue for symmetric rational functions in three variables?
Reference Symmetric functions of two noncommuting variables, by Jim Agler and N. J. Young, arXiv:1307.1588
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