Graph products of completely positive maps
Introduction Main result Consequences
Graph products of completely positive maps Scott Atkinson Vanderbilt University
ECOAS 2017 S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Fix Γ = (V , E ) a simplicial (undirected, no single-vertex loops, at most one edge between vertices) graph. Assign an algebra Av to each vertex v ∈ V . If (v , w ) ∈ E then Av and Aw commute. If (v , w ) ∈ / E then Av and Aw have no relations. Edgeless graphs ; free products Complete graphs ; tensor products S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Graph products of groups Definition For each v ∈ V let Gv be a group. The graph product FΓ Gv is given by the free product ∗v ∈V Gv modulo the relations [g , h] = 1 whenever g ∈ Gv , h ∈ Gw , and (v , w ) ∈ E .
Example Right-angled Artin groups (FΓ Z) [Baudisch ’81] Right-angled Coxeter groups (FΓ (Z/2Z)) [Chiswell ’86]
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
In operator algebras: the product of many names
[Mlotkowski ’04]: Λ-free probability
[Caspers-Fima (preprint ’14) ’17]: Graph products of operator algebras
[Speicher-Wysocza´ nski ’16]: Mixtures of classical and free independence
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Universal graph product of C ∗ -algebras Definition For each v ∈ V let Av be a unital C ∗ -algebra. The universal graph product of {Av }v ∈V is the unique unital C ∗ -algebra FΓ Av satisfying the following universal property. 1
2
There exist unital ∗-homomorphisms ιv : Av → FΓ Av such that ιv (a)ιw (b) = ιw (b)ιv (a) whenever a ∈ Av , b ∈ Aw , (v , w ) ∈ E . For any unital C ∗ -algebra B with unital ∗-homomorphisms fv : Av → B such that fv (a)fw (b) = fw (b)fv (a) whenever a, b, v , w are as above, there exists a unique unital ∗-homomorphism FΓ fv : FΓ Av → B satisfying FΓ fv ◦ ιv0 = fv0 for every v0 ∈ V . S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Reduced words (of vertices) Bookkeeping is done by considering words with letters from V . We encode the commuting relations given by Γ with the equivalence relation on words generated by the following relations. (v1 , . . . , vi , vi+1 , . . . , vn ) ∼ (v1 , . . . , vi , vi+2 , . . . , vn )
if
vi = vi+1
(v1 , . . . , vi , vi+1 , . . . , vn ) ∼ (v1 , . . . , vi+1 , vi , . . . , vn )
if
(vi , vi+1 ) ∈ E .
Definition A word v = (v1 , . . . , vn ) is reduced if whenever vk = vl , k < l, there exists a p with k < p < l such that (vk , vp ) ∈ / E.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Reduced words (of elements) ˚v = ker(ϕv ). For each v ∈ V fix ϕv ∈ S(Av ) and let A Definition A reduced word in FΓ Av is an element a ∈ FΓ Av of the form ˚v and va = (v1 , . . . , vn ) is reduced. a = a1 · · · an where ak ∈ A k Scalar multiples of the unit are reduced by convention. Let Wred denote the set of reduced words of either vertices or elements–context will tell. Fact The linear span of Wred is dense in FΓ Av .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Completely positive maps
Definition A map θ : A → B between C ∗ -algebras is completely positive if θ(n) : Mn (A) → Mn (B) (aij )ij 7→ (θ(aij ))ij is positive for every n ∈ N.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Graph product of ucp maps For each v ∈ V , let θv : Av → B be a unital completely positive (ucp) map. Densely define FΓ θv : FΓ Av → B as follows. ˚v , then put If a = a1 · · · an ∈ Wred with ak ∈ A k FΓ θv (a) = θv1 (a1 ) · · · θvn (an ) and extend linearly.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Main result
Theorem FΓ θv is ucp.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Reductions Write Θ for FΓ θv , and assume B ⊂ B(H). Suffices to show that for any n ∈ N, x1 , . . . , xn ∈ FΓ Av , ξ1 , . . . , ξn ∈ H, n X
hΘ(xi∗ xj )ξj |ξi i ≥ 0.
i,j=1
Can take x1 , . . . , xn ∈ Wred .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Complete sets Definition A finite subset X ⊂ Wred is complete if it is closed under truncation. This naturally gives a partial order with respect to truncation. Given a finite subset Y ⊂ Wred , let Y denote the completion of Y . Final reduction: Show that for any complete set X ⊂ Wred and any function ξ : X → H, we have X hΘ(x ∗ y )ξ(y )|ξ(x)i ≥ 0. (1) x,y ∈X
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Non-commutative length Definition Fix v0 ∈ V . Let v = (v1 , . . . , vn , v0 ) be reduced. We let ..v..v0 denote the (right-hand) non-commutative length of v with respect to v0 , given by .. .. v.v0 := Card {i|1 ≤ i ≤ n, (vi , v0 ) ∈ / E} . If v cannot be written with v0 at the right-hand end, put .. .. . . . . v.v0 = −1. If w ∈ FΓ Av is reduced, let .w .v0 = .vw .v0 . Given a finite set X of reduced words (of vertices or algebra elements), we define the (right-hand) non-commutative length of X with respect to v0 , denoted ..X ..v0 to be given by ..X .. := max ..w .. . v0 v0 w ∈X
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Standard form Definition Fix v0 ∈ V . Let x ∈ Wred be such that v0 ∈ x. Suppose y, c, b ∈ Wred , satisfy the following properties. x = yc(v0 )b; b is the word of smallest length so that yc(v0 ) x and .. . . . yc(v0 )..v0 = . {x} .v0 ; y is the word of smallest length so that y(v0 ) x and ..y(v ).. = .. {x} .. . 0 v0 v0 Then we say that x = yc(v0 )b is in standard form with respect to v0 . We extend this definition to reduced words of algebra elements.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Proceed by induction on |X |. Base case: |X | = 1: trivial. |X | ≥ 2: Let (v0 ) ∈ vX . Put n o . . . . X1 := x ∈ X . {x} .v0 = .X .v0 , and let x0 ∈ X1 be an element of maximal length in X1 . Say ˚v0 ). x0 = y0 c0 a0 b0 is in standard form with respect to v0 (a0 ∈ A Define n o ˚v0 ), vy = vy0 . Y1 := x ∈ X1 in standard form x = ycab (a ∈ A
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Consider the following decomposition. X X hΘ(w ∗ z)ξ(z)|ξ(w )i hΘ(x ∗ y )ξ(y )|ξ(x)i = x,y ∈X
w ,z∈X \Y1
+
X
hΘ(x ∗ x 0 )ξ(x 0 )|ξ(x)i
x,x 0 ∈Y1
X
+
2RehΘ(x ∗ z)ξ(z)|ξ(x)i.
x∈Y1 ,z∈X \Y1
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Cross-terms lemma ˚v0 , be in standard form with respect to Let x = ycab ∈ Wred , a ∈ A v0 . Then Θ(x ∗ z) = Θ(b ∗ a∗ c ∗ y ∗ z) = Θ(b ∗ a∗ )Θ(c ∗ y ∗ z) whenever z ∈ Wred satisfies either of the following conditions. . . . . 1 . {z} . v0 < . {x} .v0 . . . . 0 0 0 0 2 . {z} . v0 = . {x} .v0 but vy 6= vy 0 where z = y c a b is in standard form with respect to v0 .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Stinespring construction for concatenation Consider C|X | with standard basis {ex }x∈X . (1) yields the following positive semi-definite sesquilinear form on H ⊗ C|X | given by hξ ⊗ ey |η ⊗ ex i = hΘ(x ∗ y )ξ|ηi.
Form the resulting Hilbert space H ⊗Θ C|X | .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Stinespring construction for concatenation For each x ∈ X let Vx : H → H ⊗Θ C|X | be given by Vx (ξ) = ξ ⊗Θ ex .
Given x ∈ X with |x| = 1, define Lx : H ⊗Θ C|X | → H ⊗Θ C|X | as follows. if xy ∈ /X 0 Lx (ξ ⊗Θ ey ) = ξ ⊗Θ exy if xy ∈ X
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Proposition Lx is bounded. Yields Proposition Let X ⊂ Wred be a complete set and assume that for every ξ : X → H, (1) holds. Additionally assume that for ci , bi , ci bi ∈ X , Θ(bi∗ ci∗ cj ) = Θ(bi∗ )Θ(ci∗ cj ). Then we have the following inequality. [Θ(bi∗ ci∗ cj bj )]ij ≥ [Θ(bi∗ )Θ(ci∗ cj )Θ(bj )]ij
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
The previous proposition can be bootstrapped to obtain the following lemma. Y1 -square lemma N Let {xi }N i=1 ∈ (Wred ) be a finite sequence such that for every 1 ≤ i ≤ N, we have v0 ∈ vxi . For each 1 ≤ i ≤ N, let xi = yi ci ai bi ˚v0 ). Assume the be in standard form with respect to v0 (ai ∈ A following. 1
2
For every 1 ≤ i, j ≤ N, vyi = vyj ; For every complete set X ( ({xi }N i=1 ) and any function ξ : X → H, (1) holds.
Then [Θ(xi∗ xj )]ij ≥ [Θ(bi∗ ai∗ )Θ(ci∗ yi∗ yj cj )Θ(aj bj )]ij . S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
The inductive hypothesis gives the following. X
hΘ(w ∗ z)ξ(z)|ξ(w )i =
w ,z∈X \Y1
X
hVw∗ Vz ξ(z)|ξ(w )i
w ,z∈X \Y1
=
X
hVz ξ(z)|Vw ξ(w )i
w ,z∈X \Y1
X 2 = Vw ξ(w ) w ∈X \Y1
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
The cross-terms lemma gives the following. X
2RehΘ(x ∗ z)ξ(z)|ξ(x)i
x∈Y1 ,z∈X \Y1
=
X
2RehΘ(b ∗ a∗ )Θ(c ∗ y ∗ z)ξ(z)|ξ(ycab)i
ycab∈Y1 ,z∈X \Y1
=
X
2RehVz ξ(z)|Vyc Θ(ab)ξ(ycab)i.
ycab∈Y1 ,z∈X \Y1
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
The Y1 -square lemma gives the following. X
hΘ(x ∗ x 0 )ξ(x 0 )|ξ(x)i
x,x 0 ∈Y1
≥
X
hΘ(b ∗ a∗ )Θ(c ∗ y ∗ y 0 c 0 )Θ(a0 b 0 )ξ(y 0 c 0 a0 b 0 )|ξ(ycab)i
x=ycab,x 0 =y 0 c 0 a0 b 0 ∈Y1
=
X
hVy 0 c 0 Θ(a0 b 0 )ξ(y 0 c 0 a0 b 0 )|Vyc Θ(ab)ξ(ycab)i
ycab,y 0 c 0 a0 b 0 ∈Y1
X 2 = Vyc Θ(ab)ξ(ycab) . ycab∈Y1
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Positive-definite functions
Definition Let G be a group and H be a Hilbert space. A function f : G → B(H) is positive-definite if for every finite subset {g1 , . . . , gn } ⊂ G , the n × n matrix f (gi−1 gj ) ij is positive.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Positive-definite functions ↔ ucp maps
There is a 1-1 correspondence between positive-definite functions f : G → B(H), f (e) = 1 and ucp maps θ : C ∗ (G ) → B(H) in the following sense. If ug ∈ C ∗ (G ) denotes the unitary corresponding to the group element g ∈ G , then f → θf (ug ) := f (g ) fθ (g ) := θ(ug ) ← θ.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Graph product of positive-definite functions Definition Let H be a Hilbert space, and for each v ∈ V , let Gv be a group and fv : Gv → B(H) be positive-definite with fv (e) = 1. If (v , w ) ∈ E ⇒ fv (Gv ) and fw (Gw ) commute, then we define the graph product of the fv ’s, FΓ fv : FΓ Gv → B(H), as follows. 1
FΓ fv (e) = 1;
2
if for 1 ≤ k ≤ n, gk ∈ Gvk \ {1} and (v1 , . . . , vn ) ∈ Wred , then FΓ fv (g1 · · · gn ) := fv1 (g1 ) · · · fvn (gn ).
Theorem Let Gv , fv and H be as above. Then FΓ fv is positive-definite. S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Unitary dilation consequence
Theorem Let H be a Hilbert space, and let {Tv }v ∈V ⊂ B(H) be contractions such that if (v , w ) ∈ E then Tv and Tw doubly commute ([Tv , Tw ] = [Tv∗ , Tw ] = 0). Then there exist a Hilbert space K containing H and unitaries Uv ∈ B(K) for each v ∈ V such that for any polynomial p ∈ ChXv iv ∈V in |V | non-commuting indeterminates we have p({Tv }v ∈V ) = PH p({Uv }v ∈V )|H .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Proof. Goal: obtain a ucp map Θ : FΓ C ∗ (Z) → B(H) such that Θ(p((xv )) = p((Tv )). Define the ucp map θv on the v th copy of C ∗ (Z) as follows. m Tv if m ≥ 0 θv (xvm ) = (Tv∗ )−m if m < 0 Then the map Θ = FΓ θv : FΓ C ∗ (Z) = C ∗ (FΓ Z) → B(H) defined with respect to the canonical trace on C ∗ (Z) does the job.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Graph independence Definition Given a non-commutative probability space (A, ϕ), let {Av }v ∈V ⊂ A be a family of unital C ∗ -subalgebras. An element a ∈ C ∗ (∪v ∈V Av ) is reduced with respect to ϕ if a = a1 · · · am ˚v for 1 ≤ j ≤ m and (v1 , . . . , vm ) is reduced. where aj ∈ A j Definition Given a non-commutative probability space (A, ϕ), a family of unital C ∗ -subalgebras {Av }v ∈V ⊂ (A, ϕ) is Γ independent if 1 2
(v , v 0 ) ∈ E ⇒ Av and Av 0 commute; for any a ∈ C ∗ (∪v ∈V Av ) such that a is reduced with respect to ϕ, ϕ(a) = 0. S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Unitary dilation of graph independent contractions Theorem Given Γ independent contractions {Tv }v ∈V in the noncommutative probability space (B(H), ϕ), there exist a Hilbert space K containing H and unitaries {Uv }v ∈V ⊂ B(K) that are Γ independent with respect to ϕ ◦ Ad(PH ) such that for any polynomial p ∈ ChXv iv ∈V in |V | non-commuting indeterminates we have p({Tv }v ∈V ) = PH p({Uv }v ∈V )|H . Furthermore, this dilation is unique up to unitary equivalence if K is minimal.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
von Neumann’s inequality Theorem Let Γ = (V , E ) be a graph. Let H be a Hilbert space and {Tv }v ∈V ⊂ B(H) be contractions such that if (v , v 0 ) ∈ E then Tv and Tv 0 doubly commute ([Tv , Tv 0 ] = [Tv∗ , Tv 0 ] = 0). Let p ∈ ChXv iv ∈V be a polynomial in |V | non-commuting indeterminates. Then ||p({Tv }v ∈V )|| ≤ ||p({xv }v ∈V )||C ∗ (FΓ Z) where for each v ∈ V , xv denotes the unitary corresponding to the canonical generator of the v th copy of Z.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Thanks!
Preprint: arXiv:1706.07389
S. Atkinson
Graph products of completely positive maps
Graph products of completely positive maps Scott Atkinson Vanderbilt University
ECOAS 2017 S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Fix Γ = (V , E ) a simplicial (undirected, no single-vertex loops, at most one edge between vertices) graph. Assign an algebra Av to each vertex v ∈ V . If (v , w ) ∈ E then Av and Aw commute. If (v , w ) ∈ / E then Av and Aw have no relations. Edgeless graphs ; free products Complete graphs ; tensor products S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Graph products of groups Definition For each v ∈ V let Gv be a group. The graph product FΓ Gv is given by the free product ∗v ∈V Gv modulo the relations [g , h] = 1 whenever g ∈ Gv , h ∈ Gw , and (v , w ) ∈ E .
Example Right-angled Artin groups (FΓ Z) [Baudisch ’81] Right-angled Coxeter groups (FΓ (Z/2Z)) [Chiswell ’86]
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
In operator algebras: the product of many names
[Mlotkowski ’04]: Λ-free probability
[Caspers-Fima (preprint ’14) ’17]: Graph products of operator algebras
[Speicher-Wysocza´ nski ’16]: Mixtures of classical and free independence
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Universal graph product of C ∗ -algebras Definition For each v ∈ V let Av be a unital C ∗ -algebra. The universal graph product of {Av }v ∈V is the unique unital C ∗ -algebra FΓ Av satisfying the following universal property. 1
2
There exist unital ∗-homomorphisms ιv : Av → FΓ Av such that ιv (a)ιw (b) = ιw (b)ιv (a) whenever a ∈ Av , b ∈ Aw , (v , w ) ∈ E . For any unital C ∗ -algebra B with unital ∗-homomorphisms fv : Av → B such that fv (a)fw (b) = fw (b)fv (a) whenever a, b, v , w are as above, there exists a unique unital ∗-homomorphism FΓ fv : FΓ Av → B satisfying FΓ fv ◦ ιv0 = fv0 for every v0 ∈ V . S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Reduced words (of vertices) Bookkeeping is done by considering words with letters from V . We encode the commuting relations given by Γ with the equivalence relation on words generated by the following relations. (v1 , . . . , vi , vi+1 , . . . , vn ) ∼ (v1 , . . . , vi , vi+2 , . . . , vn )
if
vi = vi+1
(v1 , . . . , vi , vi+1 , . . . , vn ) ∼ (v1 , . . . , vi+1 , vi , . . . , vn )
if
(vi , vi+1 ) ∈ E .
Definition A word v = (v1 , . . . , vn ) is reduced if whenever vk = vl , k < l, there exists a p with k < p < l such that (vk , vp ) ∈ / E.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Reduced words (of elements) ˚v = ker(ϕv ). For each v ∈ V fix ϕv ∈ S(Av ) and let A Definition A reduced word in FΓ Av is an element a ∈ FΓ Av of the form ˚v and va = (v1 , . . . , vn ) is reduced. a = a1 · · · an where ak ∈ A k Scalar multiples of the unit are reduced by convention. Let Wred denote the set of reduced words of either vertices or elements–context will tell. Fact The linear span of Wred is dense in FΓ Av .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Completely positive maps
Definition A map θ : A → B between C ∗ -algebras is completely positive if θ(n) : Mn (A) → Mn (B) (aij )ij 7→ (θ(aij ))ij is positive for every n ∈ N.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Graph product of ucp maps For each v ∈ V , let θv : Av → B be a unital completely positive (ucp) map. Densely define FΓ θv : FΓ Av → B as follows. ˚v , then put If a = a1 · · · an ∈ Wred with ak ∈ A k FΓ θv (a) = θv1 (a1 ) · · · θvn (an ) and extend linearly.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Main result
Theorem FΓ θv is ucp.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Reductions Write Θ for FΓ θv , and assume B ⊂ B(H). Suffices to show that for any n ∈ N, x1 , . . . , xn ∈ FΓ Av , ξ1 , . . . , ξn ∈ H, n X
hΘ(xi∗ xj )ξj |ξi i ≥ 0.
i,j=1
Can take x1 , . . . , xn ∈ Wred .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Complete sets Definition A finite subset X ⊂ Wred is complete if it is closed under truncation. This naturally gives a partial order with respect to truncation. Given a finite subset Y ⊂ Wred , let Y denote the completion of Y . Final reduction: Show that for any complete set X ⊂ Wred and any function ξ : X → H, we have X hΘ(x ∗ y )ξ(y )|ξ(x)i ≥ 0. (1) x,y ∈X
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Non-commutative length Definition Fix v0 ∈ V . Let v = (v1 , . . . , vn , v0 ) be reduced. We let ..v..v0 denote the (right-hand) non-commutative length of v with respect to v0 , given by .. .. v.v0 := Card {i|1 ≤ i ≤ n, (vi , v0 ) ∈ / E} . If v cannot be written with v0 at the right-hand end, put .. .. . . . . v.v0 = −1. If w ∈ FΓ Av is reduced, let .w .v0 = .vw .v0 . Given a finite set X of reduced words (of vertices or algebra elements), we define the (right-hand) non-commutative length of X with respect to v0 , denoted ..X ..v0 to be given by ..X .. := max ..w .. . v0 v0 w ∈X
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Standard form Definition Fix v0 ∈ V . Let x ∈ Wred be such that v0 ∈ x. Suppose y, c, b ∈ Wred , satisfy the following properties. x = yc(v0 )b; b is the word of smallest length so that yc(v0 ) x and .. . . . yc(v0 )..v0 = . {x} .v0 ; y is the word of smallest length so that y(v0 ) x and ..y(v ).. = .. {x} .. . 0 v0 v0 Then we say that x = yc(v0 )b is in standard form with respect to v0 . We extend this definition to reduced words of algebra elements.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Proceed by induction on |X |. Base case: |X | = 1: trivial. |X | ≥ 2: Let (v0 ) ∈ vX . Put n o . . . . X1 := x ∈ X . {x} .v0 = .X .v0 , and let x0 ∈ X1 be an element of maximal length in X1 . Say ˚v0 ). x0 = y0 c0 a0 b0 is in standard form with respect to v0 (a0 ∈ A Define n o ˚v0 ), vy = vy0 . Y1 := x ∈ X1 in standard form x = ycab (a ∈ A
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Consider the following decomposition. X X hΘ(w ∗ z)ξ(z)|ξ(w )i hΘ(x ∗ y )ξ(y )|ξ(x)i = x,y ∈X
w ,z∈X \Y1
+
X
hΘ(x ∗ x 0 )ξ(x 0 )|ξ(x)i
x,x 0 ∈Y1
X
+
2RehΘ(x ∗ z)ξ(z)|ξ(x)i.
x∈Y1 ,z∈X \Y1
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Cross-terms lemma ˚v0 , be in standard form with respect to Let x = ycab ∈ Wred , a ∈ A v0 . Then Θ(x ∗ z) = Θ(b ∗ a∗ c ∗ y ∗ z) = Θ(b ∗ a∗ )Θ(c ∗ y ∗ z) whenever z ∈ Wred satisfies either of the following conditions. . . . . 1 . {z} . v0 < . {x} .v0 . . . . 0 0 0 0 2 . {z} . v0 = . {x} .v0 but vy 6= vy 0 where z = y c a b is in standard form with respect to v0 .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Stinespring construction for concatenation Consider C|X | with standard basis {ex }x∈X . (1) yields the following positive semi-definite sesquilinear form on H ⊗ C|X | given by hξ ⊗ ey |η ⊗ ex i = hΘ(x ∗ y )ξ|ηi.
Form the resulting Hilbert space H ⊗Θ C|X | .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Stinespring construction for concatenation For each x ∈ X let Vx : H → H ⊗Θ C|X | be given by Vx (ξ) = ξ ⊗Θ ex .
Given x ∈ X with |x| = 1, define Lx : H ⊗Θ C|X | → H ⊗Θ C|X | as follows. if xy ∈ /X 0 Lx (ξ ⊗Θ ey ) = ξ ⊗Θ exy if xy ∈ X
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Proposition Lx is bounded. Yields Proposition Let X ⊂ Wred be a complete set and assume that for every ξ : X → H, (1) holds. Additionally assume that for ci , bi , ci bi ∈ X , Θ(bi∗ ci∗ cj ) = Θ(bi∗ )Θ(ci∗ cj ). Then we have the following inequality. [Θ(bi∗ ci∗ cj bj )]ij ≥ [Θ(bi∗ )Θ(ci∗ cj )Θ(bj )]ij
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
The previous proposition can be bootstrapped to obtain the following lemma. Y1 -square lemma N Let {xi }N i=1 ∈ (Wred ) be a finite sequence such that for every 1 ≤ i ≤ N, we have v0 ∈ vxi . For each 1 ≤ i ≤ N, let xi = yi ci ai bi ˚v0 ). Assume the be in standard form with respect to v0 (ai ∈ A following. 1
2
For every 1 ≤ i, j ≤ N, vyi = vyj ; For every complete set X ( ({xi }N i=1 ) and any function ξ : X → H, (1) holds.
Then [Θ(xi∗ xj )]ij ≥ [Θ(bi∗ ai∗ )Θ(ci∗ yi∗ yj cj )Θ(aj bj )]ij . S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
The inductive hypothesis gives the following. X
hΘ(w ∗ z)ξ(z)|ξ(w )i =
w ,z∈X \Y1
X
hVw∗ Vz ξ(z)|ξ(w )i
w ,z∈X \Y1
=
X
hVz ξ(z)|Vw ξ(w )i
w ,z∈X \Y1
X 2 = Vw ξ(w ) w ∈X \Y1
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
The cross-terms lemma gives the following. X
2RehΘ(x ∗ z)ξ(z)|ξ(x)i
x∈Y1 ,z∈X \Y1
=
X
2RehΘ(b ∗ a∗ )Θ(c ∗ y ∗ z)ξ(z)|ξ(ycab)i
ycab∈Y1 ,z∈X \Y1
=
X
2RehVz ξ(z)|Vyc Θ(ab)ξ(ycab)i.
ycab∈Y1 ,z∈X \Y1
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
The Y1 -square lemma gives the following. X
hΘ(x ∗ x 0 )ξ(x 0 )|ξ(x)i
x,x 0 ∈Y1
≥
X
hΘ(b ∗ a∗ )Θ(c ∗ y ∗ y 0 c 0 )Θ(a0 b 0 )ξ(y 0 c 0 a0 b 0 )|ξ(ycab)i
x=ycab,x 0 =y 0 c 0 a0 b 0 ∈Y1
=
X
hVy 0 c 0 Θ(a0 b 0 )ξ(y 0 c 0 a0 b 0 )|Vyc Θ(ab)ξ(ycab)i
ycab,y 0 c 0 a0 b 0 ∈Y1
X 2 = Vyc Θ(ab)ξ(ycab) . ycab∈Y1
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Positive-definite functions
Definition Let G be a group and H be a Hilbert space. A function f : G → B(H) is positive-definite if for every finite subset {g1 , . . . , gn } ⊂ G , the n × n matrix f (gi−1 gj ) ij is positive.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Positive-definite functions ↔ ucp maps
There is a 1-1 correspondence between positive-definite functions f : G → B(H), f (e) = 1 and ucp maps θ : C ∗ (G ) → B(H) in the following sense. If ug ∈ C ∗ (G ) denotes the unitary corresponding to the group element g ∈ G , then f → θf (ug ) := f (g ) fθ (g ) := θ(ug ) ← θ.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Graph product of positive-definite functions Definition Let H be a Hilbert space, and for each v ∈ V , let Gv be a group and fv : Gv → B(H) be positive-definite with fv (e) = 1. If (v , w ) ∈ E ⇒ fv (Gv ) and fw (Gw ) commute, then we define the graph product of the fv ’s, FΓ fv : FΓ Gv → B(H), as follows. 1
FΓ fv (e) = 1;
2
if for 1 ≤ k ≤ n, gk ∈ Gvk \ {1} and (v1 , . . . , vn ) ∈ Wred , then FΓ fv (g1 · · · gn ) := fv1 (g1 ) · · · fvn (gn ).
Theorem Let Gv , fv and H be as above. Then FΓ fv is positive-definite. S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Unitary dilation consequence
Theorem Let H be a Hilbert space, and let {Tv }v ∈V ⊂ B(H) be contractions such that if (v , w ) ∈ E then Tv and Tw doubly commute ([Tv , Tw ] = [Tv∗ , Tw ] = 0). Then there exist a Hilbert space K containing H and unitaries Uv ∈ B(K) for each v ∈ V such that for any polynomial p ∈ ChXv iv ∈V in |V | non-commuting indeterminates we have p({Tv }v ∈V ) = PH p({Uv }v ∈V )|H .
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Proof. Goal: obtain a ucp map Θ : FΓ C ∗ (Z) → B(H) such that Θ(p((xv )) = p((Tv )). Define the ucp map θv on the v th copy of C ∗ (Z) as follows. m Tv if m ≥ 0 θv (xvm ) = (Tv∗ )−m if m < 0 Then the map Θ = FΓ θv : FΓ C ∗ (Z) = C ∗ (FΓ Z) → B(H) defined with respect to the canonical trace on C ∗ (Z) does the job.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Graph independence Definition Given a non-commutative probability space (A, ϕ), let {Av }v ∈V ⊂ A be a family of unital C ∗ -subalgebras. An element a ∈ C ∗ (∪v ∈V Av ) is reduced with respect to ϕ if a = a1 · · · am ˚v for 1 ≤ j ≤ m and (v1 , . . . , vm ) is reduced. where aj ∈ A j Definition Given a non-commutative probability space (A, ϕ), a family of unital C ∗ -subalgebras {Av }v ∈V ⊂ (A, ϕ) is Γ independent if 1 2
(v , v 0 ) ∈ E ⇒ Av and Av 0 commute; for any a ∈ C ∗ (∪v ∈V Av ) such that a is reduced with respect to ϕ, ϕ(a) = 0. S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Unitary dilation of graph independent contractions Theorem Given Γ independent contractions {Tv }v ∈V in the noncommutative probability space (B(H), ϕ), there exist a Hilbert space K containing H and unitaries {Uv }v ∈V ⊂ B(K) that are Γ independent with respect to ϕ ◦ Ad(PH ) such that for any polynomial p ∈ ChXv iv ∈V in |V | non-commuting indeterminates we have p({Tv }v ∈V ) = PH p({Uv }v ∈V )|H . Furthermore, this dilation is unique up to unitary equivalence if K is minimal.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
von Neumann’s inequality Theorem Let Γ = (V , E ) be a graph. Let H be a Hilbert space and {Tv }v ∈V ⊂ B(H) be contractions such that if (v , v 0 ) ∈ E then Tv and Tv 0 doubly commute ([Tv , Tv 0 ] = [Tv∗ , Tv 0 ] = 0). Let p ∈ ChXv iv ∈V be a polynomial in |V | non-commuting indeterminates. Then ||p({Tv }v ∈V )|| ≤ ||p({xv }v ∈V )||C ∗ (FΓ Z) where for each v ∈ V , xv denotes the unitary corresponding to the canonical generator of the v th copy of Z.
S. Atkinson
Graph products of completely positive maps
Introduction Main result Consequences
Positive-definite functions Unitary dilation von Neumann’s inequality
Thanks!
Preprint: arXiv:1706.07389
S. Atkinson
Graph products of completely positive maps
