Download Completely positive multipliers

Completely positive multipliers
Matthew Daws
Leeds
Lille, Oct 2012
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
1 / 21
Lille, Oct 2012
2 / 21
Outline
1
Locally compact quantum groups
2
Multipliers of convolution algebras
3
Bistochastic channels
Matthew Daws (Leeds)
CP Multipliers
The von Neumann algebra of a group
Let
G
=⇒
L∞ (G )
be a locally compact group
form the von Neumann algebra
Have lost the product of
injective, normal
G.
∆
L2 (G ).
∆(F )(s , t ) = F (st ).
gives the usual convolution product on
L1 (G ) ⊗ L1 (G ) → L1 (G );
is implemented by a unitary operator
F ∈ L∞ (G )
L1 (G )
ω ⊗ τ 7→ (ω ⊗ τ) ◦ ∆ = ω ? τ.
W ξ(s , t ) = ξ(s , s −1 t ),
where
can
∗-homomorphism
The pre-adjoint of
∆
acting on
=⇒
We recapture this by considering the
∆ : L∞ (G ) → L∞ (G × G );
The map
has a Haar measure
W
on
L2 (G × G ),
∆(F ) = W ∗ (1 ⊗ F )W ,
identied with the operator of multiplication by
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
F.
3 / 21
The other von Neumann algebra of a group
Let
VN (G )
be the von Neumann algebra acting on
by the left translation operators
The predual of
VN (G )
L2 (G )
generated
λs , s ∈ G .
is the Fourier algebra (a la Eymard)
A(G ),
a
commutative Banach algebra.
∆ : VN (G ) → VN (G )⊗VN (G )
product on A(G ),
Again, there is
induces the
whose pre-adjoint
∆(λs ) = λs ⊗ λs .
That such a
∆
exists follows as
^ ∗ (1 ⊗ x )W
^
∆(x ) = W
where
σ : L2 (G × G ) → L2 (G × G )
Matthew Daws (Leeds)
with
^ = σW ∗ σ,
W
is the swap map.
CP Multipliers
Lille, Oct 2012
4 / 21
How
W
governs everything
The unitary
W
W12 W13 W23 = W23 W12 ;
is multiplicative;
L∞ (G )⊗VN (G ).
and lives in
The map
L1 (G ) → VN (G );
L1 (G )
is the usual representation of
image is
σ-weakly
The map
dense in
wish). The image is
on
VN (G ),
A(G ) → L∞ (G );
is the usual embedding of
ω 7→ (ω ⊗ ι)(W ),
A(G )
σ-weakly
L2 (G )
by convolution. The
and norm dense in
Cr∗ (G ).
ω 7→ (ι ⊗ ω)(W ),
into
L∞ (G )
(the Gelfand map, if you
L∞ (G ),
dense in
and norm dense in
C0 (G ).
The group inverse is represented by the antipode
S : L∞ (G ) → L∞ (G ); S (F )(t ) = F (t −1 ).
We have that
S (ι ⊗ ω)(W ) = (ι ⊗ ω)(W ∗ ).
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
5 / 21
More non-commutative framework
M
a von Neumann algebra;
∆ a normal injective ∗-homomorphism M → M ⊗M
(∆ ⊗ ι)∆ = (ι ⊗ ∆)∆;
a left-invariant weight
ϕ,
with
(ι ⊗ ϕ)∆(·) = ϕ(·)1
with
(in some loose
sense).
a right-invariant weight
Let
H
be the GNS space for
ψ,
ϕ.
^ = σW ∗ σ.
W
(ψ ⊗ ι)∆(·) = ψ(·)1.
There is a unitary
∆(x ) = W ∗ (1 ⊗ x )W ,
Again form
with
M =
lin{(ι
on
⊗ ω)(W )}
H ⊗H
σ-weak
with
.
Then
^ =
M
lin{(ι
^ )}σ-weak
⊗ ω)(W
^ =W
^ ∗ (1 ⊗ ·)W
^
∆(·)
^.
weights ϕ
^ and ψ
is a von Neumann algebra, we can dene
coassociative. It is possible to dene
Matthew Daws (Leeds)
W
CP Multipliers
which is
Lille, Oct 2012
6 / 21
Antipode not bounded
Can again dene
S (ι ⊗ ω)(W ) = (ι ⊗ ω)(W ∗ ).
However, in general
We can factor
R
S
S
will be an unbounded,
as
σ-weakly-closed
operator.
S = R ◦ τ−i /2 .
is the unitary antipode, a normal anti-∗-homomorphism
M →M
which is an anti-homomorphism on
M∗
S ◦ ∗ ◦ S ◦ ∗ = ι.
τ−i /2
group
is the analytic generator of a one-parameter automorphism
(τt )
of
M.
Each
τt
also induces a homomorphism on
Via Tomita-Takesaki theory, the weight
^.
∇
Then
^ it (·)∇
^ −it . (τt )
τt (·) = ∇
Matthew Daws (Leeds)
ϕ
^
M∗ .
has a modular operator
is the scaling group.
CP Multipliers
Lille, Oct 2012
7 / 21
A few names
An incomplete list. . .
This viewpoint on
L∞ (G )
and
VN (G )
comes from Takesaki,
Tatsuuma.
Using
W
in a more general setting comes from Baaj, Skandalis.
Work of Woronowicz on the compact case.
Enock & Schwartz, Kac & Vainerman developed Kac algebras
(essentially when
S = R ).
Masuda, Nakagami, Woronowicz gave a more complicated (but
equivalent) set of axioms
Current axioms are from Kustermans, Vaes.
Prioritising
W
leads to the notion of a manageable multiplicative
unitary from Woronowicz (and Soªtan).
Various more algebraic approaches from van Daele.
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
8 / 21
Representation theory
A corepresentation of
(M , ∆)
(∆ ⊗ ι)(U ) = U13 U23 .
∞
If M = L (G ) and π is
is a unitary
U ∈ M ⊗B(K )
a unitary representation of
G
on
with
K,
then let
∼ L∞ (G )⊗B(K ).
U = π(t ) t ∈G ∈ L∞ (G , B(K )) =
(∆ ⊗ ι)(U ) = U13 U23 is equivalent to π(st ) = π(s )π(t ).
∗
−1 ) becomes reected in the general fact
relation π(s ) = π(s
That
The
(ι ⊗ ω)(U ) ∈ D (S ),
W
S (ι ⊗ ω)(U ) = (ι ⊗ ω)(U ∗ ).
is a corepresentation the left regular corepresentation on
Matthew Daws (Leeds)
that
CP Multipliers
H.
Lille, Oct 2012
9 / 21
∗
Reduced and universal C -algebras
{(ι ⊗ ω)(W )} gives a C∗ -algebra A. Then ∆
A → A ⊗ A (a non-denegenerate ∗-homomorphism
Taking the norm closure of
gives a morphism
A → M (A ⊗ A)).
The weights restrict to densely dened KMS weights.
∗
There is a parallel C -algebraic theory, though the axioms are more
subtle.
There is a maximal corepresentation
W
(formed from a suitable
direct sum argument). Then
^u =
A
closure{(ω
⊗ ι)(W)}
∗
is a C -algebra, which also admits a coproduct and invariant weights
(thought these might fail to be faithful).
Any corepresentation
^ u → B(K )
φ:A
U
is of the form
U = (ι ⊗ φ)(W)
where
∗-representation.
Cr∗ (G ).
is a unique non-degenerate
This parallels the formation of
Matthew Daws (Leeds)
C ∗ (G )
vs
CP Multipliers
Lille, Oct 2012
10 / 21
Multipliers
I'm interested in the algebra
said to be
M∗ ,
but this is only unital when
is
discrete.
So you can study the multipliers of
Turn
(M , ∆)
A∗
M∗ .
into a Banach algebra by using
∆
(analogue of the
measures on a group).
Then
M∗
Indeed, the same is true for
In the commutative case,
of
M∗ = L1 (G )
A∗u .
A = Au
and you get all the multipliers
as measures.
In the cocommutative case,
B (G ),
A∗ .
is an essential ideal in
A∗u
is the Fourier-Stieltjes algebra
but you get all multipliers of
M∗ = A(G )
if and only if
G
Lille, Oct 2012
11 / 21
is amenable (Bo»ejko, Losert, Nebbia).
Matthew Daws (Leeds)
CP Multipliers
Completely positive case
Suppose that
a
is a completely positive multiplier of
To be precise, multiplication by
So the adjoint is a map on
a
A(G ).
induces a map
VN (G ),
A(G ) → A(G ).
and we ask that this is
completely positive in the usual way.
(Gilbert) There is a continuous map
a (t −1 s ) = (α(t )|α(s ))K .
α:G →K
(de Canniere, Haagerup) Now immediate that
a
with
is positive denite
(and conversely).
Notice that
a
is then also a positive member of
positive functional on
So if
G
B (G ),
that is, a
C ∗ (G ).
amenable if and only if the span of the completely positive
multipliers equals the space of all (completely bounded)
multipliers.
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
12 / 21
Result in quantum case
Let
^∗ → M
^∗
L∗ : M
be a completely bounded (left) multiplier. So:
L∗ (ω ? τ) = L∗ (ω) ? τ;
the adjoint
^ →M
^
L = (L∗ )∗ : M
is completely bounded.
Theorem (Junge, Neufang, Ruan)
^ ∗ into M via
There is a unique x ∈ M such that, if we embed M
^ ), then L∗ is given by left multiplication by x .
ω 7→ (ω ⊗ ι)(W
Theorem (D.)
x ∈ M (A) and x ∗ ∈ D (S ) with S (x ∗ ) also inducing a left multiplier.
Picture: abstract multiplier of
a (continuous) function on
A(G )
corresponds to multiplication by
G.
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
13 / 21
Completely positive case
Theorem (D.)
Let L∗ be a left multiplier, associated to x ∈ M (A). The following
are equivalent:
^ →M
^ ).
L = (L∗ )∗ is completely positive (L : M
^ ∗u with L∗ (ω) = µ ? ω
There is a positive functional µ ∈ A
^ ∗u ), and x = (ι ⊗ µ)(W ∗ ).
(recall: M∗ ideal in A
There is a unitary corepresentation U of (M , ∆) on K , and a
positive µ ∈ B(K )∗ with x = (ι ⊗ µ)(U ∗ ), and with
L(^
x ) = (ι ⊗ µ) U (^
x ⊗ 1 )U ∗
Matthew Daws (Leeds)
CP Multipliers
^ ).
(^
x ∈M
Lille, Oct 2012
14 / 21
Link with Haagerup tensor product
So
L(^
x ) = (ι ⊗ µ)(U (^
x ⊗ 1)U ∗ ).
may assume that
µ
is a vector state
K,
orthonormal basis of
U
By adjusting the space
ωξ ,
and then taking
acts on, we
(ei )
an
dene
ai = (ι ⊗ ωξ,ei )(U ) ∈ M =⇒
∗
X
i
ai∗ x^ ai = L(^
x ).
∗
The extended (or weak ) Haagerup tensor product (Eros-Ruan,
Blecher-Smith, Haagerup (unpublished)) of
u ∈ M ⊗M : u =
So
X
i
X
i
Matthew Daws (Leeds)
xi ⊗ yi
M
X
with
i
with itself is the space
xi xi ,
∗
X
i
yi yi < ∞ .
∗
eh
ai∗ ⊗ ai ∈ M ⊗ M .
CP Multipliers
Sketch proof that CP multiplier
Lille, Oct 2012
⇒
15 / 21
corep
Actually, if we start with a CP left multiplier, then [JNR] (and a little
bookkeeping) shows that for some
L(^
x) =
P
i
a ∗ x^ a .
i i
eh
P
∗
i ∈I ai ⊗ ai ∈ M ⊗ M
we have
By applying the [JNR] construction twice, you nd that
X
i
ai ⊗ ai ⊗ 1 =
∗
X
i
∆(ai∗ )13 ∆(ai )23 .
This is enough to construct an isometry
ξ ∈ `2 (I )
U
on
H ⊗ `2 (I )
and
with
∗
∗
(∆ ⊗ ι)(U ∗ ) = U23
U13
,
So
U∗
(ι ⊗ ωξ,ei )(U ∗ ) = ai .
is a corepresentation; only remains to show that
unitary. This follows by using that we can nd the
U
(ai )
is
from the
Stinespring representation, and so we have some sort of
minimality condition, and then using
dense in
^M0
M
is linearly,
σ-weakly
B(H ).
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
16 / 21
Positive denite elements
f
Recall that a function
on
f (st
B(H ) ⊗ B(H ) is
maps on B(H ),
is positive denite if
) s ,t ∈G ×G = (ι ⊗ S )∆(f )
G × G.
is a positive kernel on
eh
−1
G
isomorphic to the space of completely bounded normal
x ⊗ y 7→ a 7→ xay
(a ∈ B(H )).
So can talk of complete positivity.
Theorem (D. & Salmi)
x ∈ M is a completely positive multiplier if and only if
eh
(ι ⊗ S )∆(x ) ∈ B(H ) ⊗ B(H ) and is completely positive.
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
17 / 21
To right multipliers
Introduce an involution
Dene
τ(·) = JK (·)∗ JK ,
JK
K
on
by
JK (ei ) = ei
(and anti-linearity).
an anti-∗-automorphism on
B(K ).
Then set
U c = (R ⊗ τ)(U ) =⇒ (∆ ⊗ ι)(U c ) = (R ⊗ R ⊗ τ)(σ ⊗ ι)(U13 U23 )
c Uc ) = Uc Uc .
= (σ ⊗ ι)(U23
13
13
23
So
So
Uc
is a unitary corepresentation.
L 0 (^
x ) = (ι ⊗ ω)(U c (^
x ⊗ 1)(U c )∗ )
The point is that
^ ◦ L0 ◦ R
^
r =R
positive right multiplier of
^ ∗,
M
is a left multiplier.
is then the adjoint of a completely
with
(L, r )
a double multiplier.
Not surprising from the viewpoint that we're multiplying the
(two-sided!) ideal
Matthew Daws (Leeds)
^∗
M
by elements of
CP Multipliers
^ ∗u .
A
But. . .
Lille, Oct 2012
18 / 21
For Kac algebras
Have
L
ai = (ι ⊗ ωξ,ei )(U ∗ ),
associated to
bi = (ι ⊗ ωξ,ei )((U c )∗ ).
Supposing that
and now
L0
associated to
S = R,
ai∗ = (ι ⊗ ωξ,ei )(U ∗ )∗ = (ι ⊗ ωei ,ξ )(U ) = R (ι ⊗ ωei ,ξ )(U ∗ )
= (ι ⊗ ωJK ξ,JK ei )((U c )∗ ) = (ι ⊗ ωξ,ei )((U c )∗ ) = bi .
(Assume
JK ξ = ξ).
So curiously,
X
i
So on all of
ai ai =
∗
X
i
B(H ),
x 7→
bi∗ bi = L 0 (1) = 1.
X
i
ai∗ xai ,
is a unital completely positive map, and a trace-preserving completely
positive map.
In Quantum Information Theorey, such maps are called bistochastic
quantum channels. There is a small amount of literature. . .
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
19 / 21
From a Haagerup tensor product perspective
There is an asymmetry in the extended Haagerup tensor product, so
the swap map
σ
Yet we nd that
eh
is unbounded on
u =
P
∗
i ai ⊗ ai
eh
M ⊗ M.
is such that both
u
and
σ(u )
are in
M ⊗ M.
Theorem (Pisier & Shlyakhtenko, Haagerup & Musat)
Let u ∈ B(K )⊗B(K ) be such that both u and σ(u ) are in
eh
B(K ) ⊗ B(K ). Then the map
B(K )∗ B(K )∗ → C;
ω1 ⊗ ω2 7→ hu , ω1 ⊗ ω2 i
is bounded for the minimal operator space tensor norm.
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
20 / 21
Curiosity; and a question
If also
L = L0
eh
^ ∗u satises µ = µ ◦ R
^ u ) then in M ⊗ M ,
µ∈A
X
X
X
u =
ai∗ ⊗ ai =
bi∗ ⊗ bi =
ai ⊗ ai∗ ,
(the
i
i
σ(u ) = u .
so we even have that
But in general,
u =
P
i xi ⊗ yi
eh
u ∈ B(K ) ⊗ B(K ), σ(u ) = u
with
P
i i i,
x ∗x
P
i i i < ∞.
u
What extra conditions on
If
u =
P
enough!
i
ci∗ ⊗ di
and
x 7→
y ∗y
does
not imply that
would give this representation?
P
i ci xdi
∗
is completely
Application: Completely bounded maps
positive,
A(G ) → VN (G )
that's
which
factor through a column or row Hilbert space.
Matthew Daws (Leeds)
CP Multipliers
Lille, Oct 2012
21 / 21