Symmetric functions of two noncommuting variables Nicholas Young - - PowerPoint PPT Presentation

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 D2 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) for some polynomial f and that f(x, 0) =

j fjxj. Choose

z =

• 1

−1

• ,

w =

• 1
• ;

then zw + wz = 0, zwz + wzw =

• −1
• ,

z + w =

• 1

1 −1

• .

Note that (z + w)2 = 12. Hence

• −1
• =
• j

fj

• 1

1 −1

j

= A12 + B

• 1

1 −1

• 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 Mn denote the space of complex n×n complex matrices. If f is a polynomial in nc indeterminates x1, . . . , xd then f determines a map f♯ :

∞

• n=1

(Mn)d →

∞

• n=1

Mn by substitution, as in the previous example. Poincar´ e: f = g if and only if f♯ = g♯.

Properties of polynomial nc functions

Regard the space Md def =

∞

• n=1

(Mn)d as the nc analogue of Cd. Endow Md with the discrete union topology. If f is a polynomial in d nc indeterminates then f♯ satisfies 1) f♯ maps (Mn)d to Mn for each n ≥ 1; 2) f♯(x ⊕ y) = f♯(x) ⊕ f♯(y) for all x, y ∈ Md; 3) f♯(s−1xs) = s−1f♯(x)s for all n ≥ 1 and all s ∈ Mn, , x ∈ (Mn)d with s invertible. Here s−1(x1, . . . , xd)s means (s−1x1s, . . . , s−1xds).

Analytic nc functions

• J. L. Taylor, 1974.

Let Ω ⊂ Md be an nc domain. An analytic nc function on Ω is a map f : Ω → M1 such that 1) f maps Ω ∩ (Mn)d analytically to Mn (“f is graded”); 2) f(x ⊕ y) = f(x) ⊕ f(y) for all x, y ∈ Ω (“f respects direct sums”); 3) f(s−1xs) = s−1f(x)s for all x ∈ Ω and all invertible s such that s−1xs ∈ Ω (“f respects similarities”). An nc domain is an open subset of Md that is closed under direct sums and unitary similarity.

Symmetric analytic functions on the bidisc

Let π : C2 → C2 be given by π(z, w) = (z + w, zw). If ϕ : D2 → C is analytic and symmetric in z and w then there exists an analytic function Φ : π(D2) → C such that ϕ = Φ ◦ π: D2

π

− → π(D2)

ϕ ց

ւ Φ C Are there analogous statements for symmetric functions of non-commuting variables? What is the analogue of π(D2)?

The biball B2

is the non-commutative analogue of the bidisc: B2 def =

∞

• n=1

Bn × Bn ⊂ M2 where Bn denotes the open unit ball of the space Mn of n × n complex matrices. It is an nc-domain: 1) it is open in ∞

n=1 M2 n;

2) if x = (x1, x2) ∈ B2 and y = (y1, y2) ∈ B2 then x ⊕ y def = (x1 ⊕ y1, x2 ⊕ y2) ∈ B2; 3) if x ∈ B2 ∩ M2

n and u is an n × n unitary then

(u∗x1u, u∗x2u) ∈ B2.

Symmetric analytic nc-functions on B2

There is an nc-domain Ω in the space M∞ def =

∞

• n=1

M∞

n

and an analytic nc-map π : B2 → Ω (given by a simple ratio- nal expression) such that every bounded symmetric analytic nc-function ϕ on B2 factors through π, and conversely. B2

π

− → Ω

ϕ ց

ւ Φ

• n Mn

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 : B2 → M∞ which maps x → (u, v2, vuv, vu2v, . . . ), where u = x1 + x2 2 , v = x1 − x2 2 , has the following three properties. 1) π is an analytic nc-map from B2 to Ω; 2) for every bounded symmetric analytic nc-function ϕ

• n 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 (g0, g1, g2, . . . ) ∈ M∞

n belongs to Ω if and only if

g(z) = g0 + g1z + g2z2 + . . . 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

• f D, and moreover satisfies supz∈D g(z) < 1.

Since π(x) = (u, v2, vuv, vu2v, . . . ), π(x)(z) = u + v2z + vuvz2 + vu2vz3 + . . . = u + vz(1 + uz + u2z2 + . . . )v = u + vz(1 − uz)−1v = x1 + x2 2 + x1 − x2 2 z

• 1 − x1 + x2

2 z

−1 x1 − x2

2 .

The functional calculus Let g = (g0, g1, g2, . . . ) ∈ M∞

n .

Identify g with the formal power series ∞

j=0 gjzj.

Then, for any matrix or bounded linear operator T we may write down the series ∞

j=0 gj ⊗ 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 Cn ⊗ 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 ∈ B2 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 ℓ2 and a contractive operator

• a

B C D

• : C ⊕ ℓ2 → C ⊕ ℓ2

such that the function Φ on Ω defined, for n ≥ 1 and g ∈ Ω ∩ M∞

n , by

Φ(g) = a1n + (1n ⊗ B)g(U) (1 − (1n ⊗ D)g(U))−1 (1n ⊗ C) is an nc-function satisfying ϕ = Φ ◦ π.

Another formulation Identify a sequence (gj) ∈ M∞ with the formal power series

• j≥0 gjzj. Then

π(x)(z) = x1 + x2 2 + x1 − x2 2 z

• 1 − x1 + x2

2 z

−1 x1 − x2

2 . For x ∈ B2 ∩ M2

n and a unitary operator U on ℓ2,

π(x)(U) = u ⊗ 1 + v2 ⊗ U + vuv ⊗ U2 + . . . = x1 + x2 2 ⊗ 1ℓ2 +

• x1 − x2

2 ⊗ U

• ×
• 1Cn⊗ℓ2 − x1 + x2

2 ⊗ U

−1

x1 − x2 2 ⊗ 1ℓ2

• ,

an operator on Cn ⊗ ℓ2.

Realization again Let ϕ be a symmetric nc-function on B2 bounded by 1 in

• norm. There exist a unitary U on ℓ2 and a contraction
• a

B C D

• : C ⊕ ℓ2 → C ⊕ ℓ2

such that, for x ∈ B2 ∩ M2

n,

ϕ(x) = a1n + (1n ⊗ B)π(x)(U)× (1 − (1n ⊗ D)π(x)(U))−1 (1n ⊗ C) where π(x)(U) = x1 + x2 2 ⊗ 1ℓ2 +

• x1 − x2

2 ⊗ U

• ×
• 1Cn⊗ℓ2 − x1 + x2

2 ⊗ U

−1

x1 − x2 2 ⊗ 1ℓ2

• .

A final question about symmetric polynomials The polynomials u, v2, vuv, vu2v, . . . constitute a basis for the algebra of symmetric polynomials in x1, x2, where u = 1

2(x1 + x2),

v = 1

2(x1 − x2).

Observe that vu2v = vuv(v2)−1vuv, vu3v = vuv(v2)−1vuv(v2)−1vuv, . . . Hence the algebra of rational symmetric functions in two variables has a basis consisting of the three functions u, v2, 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