Shilov Boundary for "Holomorphic Functions" on a Quantum Matrix Ball

Introduction Fock Representation Shilov Boundary

Shilov boundary for "holomorphic functions" on a quantum matrix ball

Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska)

Department of Mathematics Chalmers University of Technology

June 22, 2017

Olof Giselsson (joint work with Olga Bershtein and Daniil Proskurin, Lyudmyla Turowska) Chalmers University of Technology Shilov boundary for "holomorphic functions" on a quantum matrix ball Introduction Fock Representation Shilov Boundary

Deﬁnition of Shilov Boundary

The classical notion of a Shilov boundary

I Was introduced by Georgii Shilov.

I It generalizes the maximus modulus priciple: I If f (z) ∈ C(D) is an holomorphic function on the unit disc D, then max |f (z)| = max |f (z)|, z∈D z∈T where T is the unit circle.

Deﬁnition of Shilov Boundary

The classical notion of a Shilov boundary

I Was introduced by Georgii Shilov.

I It generalizes the maximus modulus priciple: I If f (z) ∈ C(D) is an holomorphic function on the unit disc D, then max |f (z)| = max |f (z)|, z∈D z∈T where T is the unit circle.

Deﬁnition of Shilov Boundary

The classical notion of a Shilov boundary

I Was introduced by Georgii Shilov.

I It generalizes the maximus modulus priciple: I If f (z) ∈ C(D) is an holomorphic function on the unit disc D, then max |f (z)| = max |f (z)|, z∈D z∈T where T is the unit circle.

Deﬁnition of Shilov Boundary

The classical notion of a Shilov boundary

I Was introduced by Georgii Shilov.

I It generalizes the maximus modulus priciple: I If f (z) ∈ C(D) is an holomorphic function on the unit disc D, then max |f (z)| = max |f (z)|, z∈D z∈T where T is the unit circle.

Deﬁnition of Shilov Boundary If X is a compact space and let A ⊆ C(X) be a uniform algebra. Deﬁnition A boundary of X relative to A is a closed subset B ⊆ X, such that for every f (z) ∈ A

max |f (z)| = max |f (z)|. z∈X z∈B

Deﬁnition A boundary S is called the Shilov boundary of X relative to A if it is contained in any other boundary.

It can be shown that the Shilov boundary exists and that it is unique.

Deﬁnition of Shilov Boundary If X is a compact space and let A ⊆ C(X) be a uniform algebra. Deﬁnition A boundary of X relative to A is a closed subset B ⊆ X, such that for every f (z) ∈ A

max |f (z)| = max |f (z)|. z∈X z∈B

Deﬁnition A boundary S is called the Shilov boundary of X relative to A if it is contained in any other boundary.

It can be shown that the Shilov boundary exists and that it is unique.

Deﬁnition of Shilov Boundary If X is a compact space and let A ⊆ C(X) be a uniform algebra. Deﬁnition A boundary of X relative to A is a closed subset B ⊆ X, such that for every f (z) ∈ A

max |f (z)| = max |f (z)|. z∈X z∈B

Deﬁnition A boundary S is called the Shilov boundary of X relative to A if it is contained in any other boundary.

It can be shown that the Shilov boundary exists and that it is unique.

Deﬁnition of Shilov Boundary If X is a compact space and let A ⊆ C(X) be a uniform algebra. Deﬁnition A boundary of X relative to A is a closed subset B ⊆ X, such that for every f (z) ∈ A

max |f (z)| = max |f (z)|. z∈X z∈B

Deﬁnition A boundary S is called the Shilov boundary of X relative to A if it is contained in any other boundary.

It can be shown that the Shilov boundary exists and that it is unique.

Examples of Shilov Boundary

Example In C(D) let A(D) ⊆ C(D) be the holomorphic functions. The Shilov boundary of D relative to A(D) is then T. Example Let Dn be the closed unit ball in Mn(C), then the sub-algebra A(Dn) ⊆ C(Dn) of holomorphic functions has as its Shilov boundary Un, the Lie group of unitary n × n matrices.

Examples of Shilov Boundary

Deﬁnition of Shilov Boundary

A non-commutative version of the Shilov boundary was proposed by Arveson [’70, Acta Math.]: ∗ C(X) unital C -algebras B (noncom. topological spaces) K ⊂ X, closed closed ideals in B ∗ uniform algebra subalgebra A that generates the C -algebra B, 1 ∈ A.

Deﬁnition of Shilov Boundary

Let D be a C∗-algebra and A ⊆ D a unital subalgebra that generates D as a C∗-algebra. Deﬁnition An ideal J in D is called a boundary ideal of D relative A if the canonical map j : D → D/J is a complete isometry when restricted to A. A boundary ideal is called a Shilov boundary ideal if it contains any other boundary ideal. Shilov boundary ideal exists and unique (Arveson ’70, Hamana ’79).

Deﬁnition of Shilov Boundary

The ∗-algebra Pol(Matn)q Let q be a constant 0 < q < 1. Deﬁnition j C[Matn]q is the algebra over C with generators {zk |1 ≤ k, j ≤ n} subject to the relations

α β β α za zb − qzb za = 0, a = b & α < β, or a < b & α = β, α β β α za zb − zb za = 0, α < β & a > b, α β β α −1 β α za zb − zb za − (q − q )za zb = 0, α < β & a < b.

C[Matn]q can be seen as a q-analogue of the algebra of holomorphic polynomials on Matn.

The ∗-algebra Pol(Matn)q Let q be a constant 0 < q < 1. Deﬁnition j C[Matn]q is the algebra over C with generators {zk |1 ≤ k, j ≤ n} subject to the relations

C[Matn]q can be seen as a q-analogue of the algebra of holomorphic polynomials on Matn.

The ∗-algebra Pol(Matn)q

Pol(Matn)q is a q-analog of the polynomial algebra on Matn. Its j j ∗ generators are zk , (zk ) , 1 ≤ k, j ≤ n, and the list of relations is the same as those of C[Matn]q but also that

n n β ∗ α 2 X X b0a0 β0α0 α0 β0 ∗ 2 αβ (zb ) za = q · Rba Rβα · za0 (zb0 ) + (1 − q )δabδ , a0,b0=1 α0,β0=1

αβ kl with δab, δ being the Kronecker symbols, and coefﬁcients Rij . ∼ We have Pol(Mat1)q = Pol(C)q the "quantum unit disc".

Pol(Matn)q was introduced by Vaksman & Co-authors in the 90’s.

The ∗-algebra Pol(Matn)q

Pol(Matn)q is a q-analog of the polynomial algebra on Matn. Its j j ∗ generators are zk , (zk ) , 1 ≤ k, j ≤ n, and the list of relations is the same as those of C[Matn]q but also that

n n β ∗ α 2 X X b0a0 β0α0 α0 β0 ∗ 2 αβ (zb ) za = q · Rba Rβα · za0 (zb0 ) + (1 − q )δabδ , a0,b0=1 α0,β0=1

αβ kl with δab, δ being the Kronecker symbols, and coefﬁcients Rij . ∼ We have Pol(Mat1)q = Pol(C)q the "quantum unit disc".

Pol(Matn)q was introduced by Vaksman & Co-authors in the 90’s.

The algebra C[GLn]q j The q-determinant is given by the formula for z = (zi )

X l(s) s(1) s(2) s(n) detqz = (−q) z1 z2 ... zn s∈Sn

with l(s) = card{(i, j)| i < j & s(i) > s(j)}.

It is known that detqz is in the center of C[Matn]q. Deﬁnition Let C[GLn]q be localization of C[Matn]q w.r.t the multiplicative system (detqz)N.

The algebra C[GLn]q is a q-analogue of functions on the Lie group of invertible n × n matrices.

The algebra C[GLn]q j The q-determinant is given by the formula for z = (zi )

X l(s) s(1) s(2) s(n) detqz = (−q) z1 z2 ... zn s∈Sn

with l(s) = card{(i, j)| i < j & s(i) > s(j)}.

It is known that detqz is in the center of C[Matn]q. Deﬁnition Let C[GLn]q be localization of C[Matn]q w.r.t the multiplicative system (detqz)N.

The algebra C[GLn]q is a q-analogue of functions on the Lie group of invertible n × n matrices.

The algebra C[GLn]q j The q-determinant is given by the formula for z = (zi )

X l(s) s(1) s(2) s(n) detqz = (−q) z1 z2 ... zn s∈Sn

with l(s) = card{(i, j)| i < j & s(i) > s(j)}.

It is known that detqz is in the center of C[Matn]q. Deﬁnition Let C[GLn]q be localization of C[Matn]q w.r.t the multiplicative system (detqz)N.

The algebra C[GLn]q is a q-analogue of functions on the Lie group of invertible n × n matrices.

C[Un]q and C[SUn]q

Recall that C[Un]q is C[GLn]q with the involution ∗ given by

j ∗ k+j−2n −1 j (zk ) = (−q) (detq z) detq zk

j 0 0 Where zk is z with the k th row and j th column deleted. −n(n−1)/2 C[SUn]q is the algebra C[Un]q/h(q − detqz)i. n−k j Letting the generators of C[SUn]q be tkj := q zk then this is a quantum group with co-product ∆, co-unit and antipode S

P ∗ ∆(tkj ) = m tkm ⊗ tmj (tkj ) = δkj S(tkj ) = tjk

C[Un]q and C[SUn]q

Recall that C[Un]q is C[GLn]q with the involution ∗ given by

j ∗ k+j−2n −1 j (zk ) = (−q) (detq z) detq zk

P ∗ ∆(tkj ) = m tkm ⊗ tmj (tkj ) = δkj S(tkj ) = tjk

The co-action of C[SUn]q on Pol(Matn)q

Then there is a co-action of C[SUn]q on Pol(Matn)q given by a ∗-homomorphism

D : Pol(Matn)q → Pol(Matn)q ⊗ C[SUn]q ⊗ C[SUn]q

n j X b D(zk ) = za ⊗ tak ⊗ tbj . a,b=1 In the case q = 1, this co-action comes from the two different actions of SUn on Matn

(A, X) 7→ AX (A, X) 7→ XAt .

for X ∈ Matn and A ∈ SUn.

The Fock representation πF,n of Pol(Matn)q

Vaksman et al proved the following: Theorem There exists a unique faithful bounded irreducible ∗-representation πF,n : Pol(Matn)q → B(HF,n) deﬁned by the property that there exists a vector v0 ∈ HF,n s.t

j ∗ πF,n(zk ) v0 = 0 for 1 ≤ k, j ≤ n.

The vector v0 is called a vacuum vector.

The Fock representation πF,n of Pol(Matn)q

j ∗ πF,n(zk ) v0 = 0 for 1 ≤ k, j ≤ n.

The vector v0 is called a vacuum vector.

Deﬁnition of Pol(Matn)q again

n n β ∗ α 2 X X b0a0 β0α0 α0 β0 ∗ 2 αβ (zb ) za = q · Rba Rβα · za0 (zb0 ) + (1 − q )δabδ , a0,b0=1 α0,β0=1

αβ kl with δab, δ being the Kronecker symbols, and coefﬁcients Rij .

The ∗-algebra Pol(Matn)q

For m ≤ n there exists a ∗-homomorphism

Pol(Matm)q → Pol(Matn)q

j j+n−m zk 7→ zk+n−m and especially

ρ : Pol(Matn)q → Pol(Mat2n)q

j j+n zk 7→ zk+n.

The ∗-algebra Pol(Matn)q There is a ∗-homomorphism

ψ : Pol(Matn)q → C[SUn]q

j k−n zk 7→ (−q) tkj . Hence there is a ∗-homomorphism

ψ ◦ ρ : Pol(Matn)q → C[SU2n]q

j k−n zk 7→ (−q) t(k+n),(j+n)

The idea is to construct a ∗-representation Π of C[SU2n]q s.t ∼ Π ◦ ψ ◦ ρ = πF,n.

∗-representations of C[SUn]q We present quickly parts of the representation theory of C[SUn]q due to Soibelman. 2 If π : C[SU2]q → B(` (Z+)) is the ∗-representation given by ∗ π(t11) = S Cq, π(t12) = −qDq, π(t21) = Dq, π(t22) = CqS ∞ where, in the standard orthonormal basis {em}m=0, we have

p 2m m Sem = em+1, Cqem = 1 − q em, Dqem = q em.

Consider the ∗-homomorphisms C[SUn]q → C[SU2]q

φi (tii ) = t11, φi (ti+1i+1) = t22, φi (tii+1) = t12, φi (ti+1i ) = t21 and φi (tkj ) = δk,j I otherwise.

p 2m m Sem = em+1, Cqem = 1 − q em, Dqem = q em.

Consider the ∗-homomorphisms C[SUn]q → C[SU2]q

φi (tii ) = t11, φi (ti+1i+1) = t22, φi (tii+1) = t12, φi (ti+1i ) = t21 and φi (tkj ) = δk,j I otherwise.

∗-representations of C[SUn]q

2 Let πi : C[SUn]q → B(` (Z+)) be the composition π ◦ φi . Let si denote the adjacent transposition (i, i + 1) in the symmetric group Sn. Deﬁnition For an element s ∈ Sn consider a minimal decomposition of

s = sj1 sj2 ... sjm into a product of adjacent transposition and let πs be the ∗-representation of C[SUn]q given by

πj1 ⊗ πj2 ⊗ · · · ⊗ πjm .

Up to isomorphism, πs is independent of the speciﬁc minimal decomposition. Up to action of torus, all irreducible representation of C[SUn]q arises this way (Soibelman).

Construction of Fock representation In S2n let

1 2 ... n n + 1 n + 2 ... 2n s = . n + 1 n + 2 ... 2n 1 2 ... n

Theorem πs ◦ ψ ◦ ρ is isomorphic to the Fock representation πF,n and ∗ ⊗n2 πF,n(Pol(Matn)q) ⊆ C (S) . Combining this with the faithfulness of the Fock representation gives Corollary Pol(Matn)q is isomorphic to the ∗-sub-algebra of C[SU(2n)]q generated by {t(k+n)(j+n)} with 1 ≤ k, j ≤ n.

Lifting representations of Pol(Matn)q

Theorem (G.)

For every ∗-representation π of Pol(Matn)q there is a ∗-representation Π of C[SU2n]q s.t

Π ◦ ψ ◦ ρ =∼ π.

Generalizes √ T ∗ I − T ∗T T ∈ Mat ⇒ √ ∈ SU . n − I − TT ∗ T 2n

Lifting representations of Pol(Matn)q

Theorem (G.)

For every ∗-representation π of Pol(Matn)q there is a ∗-representation Π of C[SU2n]q s.t

Π ◦ ψ ◦ ρ =∼ π.

Generalizes √ T ∗ I − T ∗T T ∈ Mat ⇒ √ ∈ SU . n − I − TT ∗ T 2n

Lifting representations of Pol(Matn)q

Theorem (G.)

For every ∗-representation π of Pol(Matn)q there is a ∗-representation Π of C[SU2n]q s.t

Π ◦ ψ ◦ ρ =∼ π.

Generalizes √ T ∗ I − T ∗T T ∈ Mat ⇒ √ ∈ SU . n − I − TT ∗ T 2n

Universality of the Fock

Theorem (G.) ∗ The universal C -algebra of Pol(Matn)q exists and is isomorphic ∗ ⊗n2 to πF,n(Pol(Matn)q) ⊆ C (S) . Every other irreducible representation is a quotient of C∗(S)⊗n2 .

∗ We denote the universal C -algebra of Pol(Matn)q by C(Dn)q

n−k j (n) Let Z = ((q) zk )k,j . Vaksman proved ||πF,n(Z )|| ≤ 1.

Universality of the Fock

Theorem (G.) ∗ The universal C -algebra of Pol(Matn)q exists and is isomorphic ∗ ⊗n2 to πF,n(Pol(Matn)q) ⊆ C (S) . Every other irreducible representation is a quotient of C∗(S)⊗n2 .

∗ We denote the universal C -algebra of Pol(Matn)q by C(Dn)q

n−k j (n) Let Z = ((q) zk )k,j . Vaksman proved ||πF,n(Z )|| ≤ 1.

Shilov Boundary

The q-analogue of holomorphic polynomials is the algebra A(Matn)q ⊆ Pol(Matn)q generated by the holomorphic j generators {zk }.

Consider the ideal Jn in Pol(Matn)q generated by the elements: ∗ n = 1: Jn = hz1z1 − 1i Pn 2n−α−β α β ∗ α,β n > 1: Jn = h j=1 q zj (zj ) − δ , α, β = 1, 2,..., ni

Jn is called the algebraic Shilov boundary ideal.

Shilov Boundary

The q-analogue of holomorphic polynomials is the algebra A(Matn)q ⊆ Pol(Matn)q generated by the holomorphic j generators {zk }.

Consider the ideal Jn in Pol(Matn)q generated by the elements: ∗ n = 1: Jn = hz1z1 − 1i Pn 2n−α−β α β ∗ α,β n > 1: Jn = h j=1 q zj (zj ) − δ , α, β = 1, 2,..., ni

Jn is called the algebraic Shilov boundary ideal.

Shilov Boundary

The q-analogue of holomorphic polynomials is the algebra A(Matn)q ⊆ Pol(Matn)q generated by the holomorphic j generators {zk }.

Consider the ideal Jn in Pol(Matn)q generated by the elements: ∗ n = 1: Jn = hz1z1 − 1i Pn 2n−α−β α β ∗ α,β n > 1: Jn = h j=1 q zj (zj ) − δ , α, β = 1, 2,..., ni

Jn is called the algebraic Shilov boundary ideal.

Shilov Boundary

The q-analogue of holomorphic polynomials is the algebra A(Matn)q ⊆ Pol(Matn)q generated by the holomorphic j generators {zk }.

Consider the ideal Jn in Pol(Matn)q generated by the elements: ∗ n = 1: Jn = hz1z1 − 1i Pn 2n−α−β α β ∗ α,β n > 1: Jn = h j=1 q zj (zj ) − δ , α, β = 1, 2,..., ni

Jn is called the algebraic Shilov boundary ideal.

Algebraic Shilov Boundary Ideal

It was proven by Vaksman that ∼ Pol(Matn)q/J = C[Un]q

i.e the name make sense.

Let A(Dn)q ⊆ C(Dn)q be the closure of A(Matn)q and ¯ Jn ⊆ C(Dn)q be the closure of Jn. ¯ Is Jn the Shilov boundary ideal for A(Dn)q relative to C(Dn)q?

Algebraic Shilov Boundary Ideal

It was proven by Vaksman that ∼ Pol(Matn)q/J = C[Un]q

i.e the name make sense.

Let A(Dn)q ⊆ C(Dn)q be the closure of A(Matn)q and ¯ Jn ⊆ C(Dn)q be the closure of Jn. ¯ Is Jn the Shilov boundary ideal for A(Dn)q relative to C(Dn)q?

Algebraic Shilov Boundary Ideal

It was proven by Vaksman that ∼ Pol(Matn)q/J = C[Un]q

i.e the name make sense.

Let A(Dn)q ⊆ C(Dn)q be the closure of A(Matn)q and ¯ Jn ⊆ C(Dn)q be the closure of Jn. ¯ Is Jn the Shilov boundary ideal for A(Dn)q relative to C(Dn)q?

Shilov Boundary

Theorem ¯ Jn is the Shilov boundary ideal of A(D)q relative to C(Dn)q We sketch the proof:

The idea is to ﬁrst prove

πF,n(a) = PH ψ(a)|H

∀a ∈ A(Dn) ¯ where the ∗-representation ψ annihilates Jn. This will show that is a boundary ideal.

Shilov Boundary

Theorem ¯ Jn is the Shilov boundary ideal of A(D)q relative to C(Dn)q We sketch the proof:

The idea is to ﬁrst prove

πF,n(a) = PH ψ(a)|H

∀a ∈ A(Dn) ¯ where the ∗-representation ψ annihilates Jn. This will show that is a boundary ideal.

Sketch of proof Theorem (Sz-Nagy) Let T ∈ B(H) with kT k ≤ 1. Then there exists a Hilbert space K,K ⊃ H and a unitary operator U on K such that

n n T = PH U |H ∀n ≥ 0.

Consider the case when n = 1, the "quantum unit disc" case 1 We have k πF,1(z1 )k ≤ 1. By (Sz-Nagy), 1 k k πF,1(z1 ) = PH U |H 1 ¯ and z1 7→ U annihilates J1. This proves the case n = 1.

The induction step In general, we use induction on n. Assume the result holds for n − 1. I For ϕ ∈ [−π, π), there are homomorphisms Πϕ : Pol(Matn)q → Pol(Mat(n−1))q I If π is a ∗-representation of Pol(Matn−1)q annihilating Jn−1, then π ◦ Πϕ annihilates Jn. I Recall the co-action

D : Pol(Matn)q → Pol(Matn)q ⊗ C[SUn]q ⊗ C[SUn]q.