Unitary Dilation of Freely Independent Contractions Scott Atkinson - - PowerPoint PPT Presentation

Unitary Dilation of Freely Independent Contractions

Scott Atkinson (University of Virginia) joint work with Chris Ramsey (U. Manitoba) YMC ∗A 2016

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Sz.-Nagy Dilation Theorem

Theorem (Sz.-Nagy) Given a contraction T ∈ B(H), there is a Hilbert space K containing H and a unitary U ∈ B(K) such that T n = PHUn|H for n ∈ Z.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Sz.-Nagy Dilation Theorem

Proof (Sch¨ affer). Let K = ℓ2(Z) ⊗ H and U =              ... ... I DT ∗ T −T ∗ DT I ... ...              where for ||A|| ≤ 1, DA = (I − A∗A)

1 2 .

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Consequence: von Neumann’s Inequality

Theorem (von Neumann) Given a contraction T ∈ B(H) and a polynomial p, we have that ||p(T)|| ≤ ||p(z)||D = ||p(z)||T. Proof. ||p(T)|| = ||PHp(U)|H|| ≤ ||p(U)|| = ||p(z)||σ(U) ≤ ||p(z)||T.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Ando’s Dilation Theorem

Theorem (Ando) Given two commuting contractions S, T ∈ B(H), there is a Hilbert space K containing H and commuting unitaries U, V ∈ B(K) such that SmT n = PHUmV n|H for n, m ∈ N. The above theorem does not hold for three or more commuting

• contractions. Counterexamples are due to Parrott and Varopoulos.
• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Sz.-Nagy-Foias Dilation Theorem

Definition Two operators S, T ∈ B(H) are said to doubly commute if ST = TS and S∗T = TS∗. Theorem (Sz.-Nagy-Foias) Given n doubly commuting contractions T1, . . . , Tn ∈ B(H), there is a Hilbert space K containing H and doubly commuting unitaries U1, . . . , Un ∈ B(K) so that T k1

1 · · · T kn n = PHUk1 1 · · · Ukn n |H

for k1, . . . , kn ∈ Z.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Non-Commutative Probability Space

Definition A non-commutative probability space is given by a pair (A, ϕ) where A is a unital C ∗-algebra and ϕ ∈ S(A) is a state on A.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Doubly Commuting Tensor Independence

Definition Let (A, ϕ) be a non-commutative probability space. The elements T1, . . . , Tn ∈ A are tensor independent or classically independent (with respect to ϕ) if

1 The generated C ∗-algebras C ∗(1, T1), . . . , C ∗(1, Tn) pair-wise

commute.

2 Given aj ∈ C ∗(1, Tj), 1 ≤ j ≤ n, we have that

ϕ(a1 · · · an) = ϕ(a1) · · · ϕ(an). Key observation: T1, . . . Tn doubly commute if and only if C ∗(1, T1), . . . , C ∗(1, Tn) pair-wise commute.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Unitary Dilation of Tensor Independent Contractions

Theorem (A.-Ramsey) Let T1, . . . , Tn be tensor independent contractions in the non-commutative probability space (B(H), ϕ). There is a Hilbert space K containing H and unitaries U1, . . . , Un ∈ B(K) that are tensor independent with respect to ψ = ϕ ◦ Ad(PH) such that T k1

1 · · · T kn n = PHUk1 1 · · · Ukn n |H

for k1, . . . , kn ∈ Z.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Free Independence

There are other notions of independence in the theory of non-commutative probability. Free independence was introduced by Voiculescu in the 1980’s and has been heavily studied ever since. Definition Let (A, ϕ) be a non-commutative probability space. The elements T1, . . . , Tn ∈ A are freely independent (with respect to ϕ) if for 1 ≤ j ≤ m, whenever aj ∈ C ∗(1, Tij) with ij = ij+1 and ϕ(aj) = 0 then we have ϕ(a1 · · · am) = 0. OR: “alternating products of centered elements are centered.”

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Free Independence

Example Let Γ be a discrete group with infinite cardinality. We say that n subgroups Γ1, . . . , Γn ≤ Γ are free from one another if for 1 ≤ j ≤ m, whenever gj ∈ Γij with ij = ij+1 and gj = e then we have g1 · · · gm = e. Consider C ∗

r (Γ) ⊂ B(ℓ2(Γ)) with state ϕ given by ϕ(·) := · δe|δe;

• bserve that g = e ⇔ ϕ(λ(g)) = 0. So Γ1, . . . , Γn ≤ Γ are free

from one another if for 1 ≤ j ≤ m, whenever gj ∈ Γij with ij = ij+1 and ϕ(λ(gj)) = 0 then we have ϕ(λ(g1) · · · λ(gm)) = 0.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Unitary Dilation of Freely Independent Contractions

Theorem (A.-Ramsey) Let T1, . . . , Tn be freely independent contractions in the non-commutative probability space (B(H), ϕ). There is a Hilbert space K containing H and unitaries U1, . . . , Un ∈ B(K) that are freely independent with respect to ψ = ϕ ◦ Ad(PH) such that T k1

i1 · · · T km im = PHUk1 i1 · · · Ukm im |H

for k1, . . . , km ∈ N.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Proof Sketch

Let Vi ∈ B(Ki) be the Sch¨ affer dilation for each Ti. Let θi : C ∗(Vi) → C ∗(1, Ti) be the ucp map given by p(Vi) → p(Ti). By Boca, there is a ucp map (coherent with ϕ) ∗n

i=1θi : ˇ

∗n

i=1C ∗(Vi) → C ∗(1, T1, . . . , Tn).

By Stinespring, let K be a Hilbert space containing H and π : ˇ ∗n

i=1C ∗(Vi) → B(K) be a ∗-homomorphism be such that

∗n

i=1θi(a) = PHπ(a)|H.

Put Ui = π(Vi) for desired unitaries.

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

More Observations

Because C ∗(Ui) is commutative for each 1 ≤ i ≤ n, ϕ ◦ Ad(PH)|C ∗(Ui) is a trace. It is a well-known fact that free independence of the Ui’s implies that ϕ ◦ Ad(PH) is a trace on C ∗(U1, . . . , Un). If ϕ is additionally faithful, then (C ∗(U1, . . . , Un), ϕ ◦ Ad(PH) ∼ = ∗n

i=1(C ∗(Vi), ϕ ◦ θi).

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Outlook

It would be nice to obtain a more concrete dilation (think Sch¨ affer). The Sz.-Nagy Unitary Dilation Theorem can be used for a slick proof of von Neumann’s inequality. Can we use free dilation to establish an inequality involving freely independent contractions? Can other forms of independence in n.c. probability be dilated?

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions

Thanks! Preprint: arXiv:1601.00613

• S. Atkinson (U.Va.)

Unitary Dilation of Freely Independent Contractions