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) YM C ∗ 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 = P H U n | H for n ∈ Z . S. Atkinson (U.Va.) Unitary Dilation of Freely Independent Contractions

Sz.-Nagy Dilation Theorem Proof (Sch¨ affer). Let K = ℓ 2 ( Z ) ⊗ H and ...   ...    0      I 0   U =   D T ∗ T     − T ∗ D T 0     0 I    ... ...  1 2 . where for || A || ≤ 1 , D A = ( I − A ∗ A ) 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 ) || = || P H p ( 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 S m T n = P H U m V 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 T 1 , . . . , T n ∈ B ( H ) , there is a Hilbert space K containing H and doubly commuting unitaries U 1 , . . . , U n ∈ B ( K ) so that T k 1 1 · · · T k n n = P H U k 1 1 · · · U k n n | H for k 1 , . . . , k n ∈ 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 T 1 , . . . , T n ∈ A are tensor independent or classically independent (with respect to ϕ ) if 1 The generated C ∗ -algebras C ∗ (1 , T 1 ) , . . . , C ∗ (1 , T n ) pair-wise commute. 2 Given a j ∈ C ∗ (1 , T j ) , 1 ≤ j ≤ n , we have that ϕ ( a 1 · · · a n ) = ϕ ( a 1 ) · · · ϕ ( a n ) . Key observation: T 1 , . . . T n doubly commute if and only if C ∗ (1 , T 1 ) , . . . , C ∗ (1 , T n ) pair-wise commute. S. Atkinson (U.Va.) Unitary Dilation of Freely Independent Contractions

Unitary Dilation of Tensor Independent Contractions Theorem (A.-Ramsey) Let T 1 , . . . , T n be tensor independent contractions in the non-commutative probability space ( B ( H ) , ϕ ) . There is a Hilbert space K containing H and unitaries U 1 , . . . , U n ∈ B ( K ) that are tensor independent with respect to ψ = ϕ ◦ Ad ( P H ) such that T k 1 n = P H U k 1 1 · · · T k n 1 · · · U k n n | H for k 1 , . . . , k n ∈ 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 T 1 , . . . , T n ∈ A are freely independent (with respect to ϕ ) if for 1 ≤ j ≤ m , whenever a j ∈ C ∗ (1 , T i j ) with i j � = i j +1 and ϕ ( a j ) = 0 then we have ϕ ( a 1 · · · a m ) = 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 Γ 1 , . . . , Γ n ≤ Γ are free from one another if for 1 ≤ j ≤ m , whenever g j ∈ Γ i j with i j � = i j +1 and g j � = e then we have g 1 · · · g m � = e . Consider C ∗ r (Γ) ⊂ B ( ℓ 2 (Γ)) with state ϕ given by ϕ ( · ) := �· δ e | δ e � ; observe that g � = e ⇔ ϕ ( λ ( g )) = 0. So Γ 1 , . . . , Γ n ≤ Γ are free from one another if for 1 ≤ j ≤ m , whenever g j ∈ Γ i j with i j � = i j +1 and ϕ ( λ ( g j )) = 0 then we have ϕ ( λ ( g 1 ) · · · λ ( g m )) = 0 . S. Atkinson (U.Va.) Unitary Dilation of Freely Independent Contractions

Unitary Dilation of Freely Independent Contractions Theorem (A.-Ramsey) Let T 1 , . . . , T n be freely independent contractions in the non-commutative probability space ( B ( H ) , ϕ ) . There is a Hilbert space K containing H and unitaries U 1 , . . . , U n ∈ B ( K ) that are freely independent with respect to ψ = ϕ ◦ Ad ( P H ) such that T k 1 i 1 · · · T k m i m = P H U k 1 i 1 · · · U k m i m | H for k 1 , . . . , k m ∈ N . S. Atkinson (U.Va.) Unitary Dilation of Freely Independent Contractions

Proof Sketch Let V i ∈ B ( K i ) be the Sch¨ affer dilation for each T i . Let θ i : C ∗ ( V i ) → C ∗ (1 , T i ) be the ucp map given by p ( V i ) �→ p ( T i ) . By Boca, there is a ucp map (coherent with ϕ ) ∗ n ∗ n i =1 θ i : ˇ i =1 C ∗ ( V i ) → C ∗ (1 , T 1 , . . . , T n ) . By Stinespring, let K be a Hilbert space containing H and ∗ n π : ˇ i =1 C ∗ ( V i ) → B ( K ) be a ∗ -homomorphism be such that ∗ n i =1 θ i ( a ) = P H π ( a ) | H . Put U i = π ( V i ) for desired unitaries. S. Atkinson (U.Va.) Unitary Dilation of Freely Independent Contractions

More Observations Because C ∗ ( U i ) is commutative for each 1 ≤ i ≤ n , ϕ ◦ Ad( P H ) | C ∗ ( U i ) is a trace. It is a well-known fact that free independence of the U i ’s implies that ϕ ◦ Ad( P H ) is a trace on C ∗ ( U 1 , . . . , U n ). If ϕ is additionally faithful, then ( C ∗ ( U 1 , . . . , U n ) , ϕ ◦ Ad( P H ) ∼ = ∗ n i =1 ( C ∗ ( V i ) , ϕ ◦ θ i ) . 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