A 1-cohomology characterization of property (T) in von Neumann algebras

0.21 MB·English

by  Jesse Peterson

A 1-COHOMOLOGY CHARACTERIZATION OF PROPERTY (T)
IN VON NEUMANN ALGEBRAS
6
0
0
2 JESSE PETERSON

n
a University of California, Los Angeles
J

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Abstract. We obtain a characterization of property (T) for von Neumann algebras in
]
A terms of 1-cohomology similar to the Delorme-Guichardet Theorem for groups.
O
.
h
t 0. Introduction.
a
m
[ The analogue of group representations in von Neumann algebras is the notion of

2 correspondences which is due to Connes ([C2], [C3], [P1]), and has been a very useful
v in defining notions such as property (T) and amenability for von Neumann algebras.
7
2 It is often useful to view group representations as positive definite functions which we
5 obtain through a GNS construction. Correspondences of a von Neumann algebra N
9
can also be viewed in two separate ways, as Hilbert N-N bimodules H, or as completely
0
4 positive maps φ : N → N, and the equivalence of these two descriptions is also realized
0
via a GNS construction. This allows one to characterize property (T) for von Neumann
/
h
algebras in terms of completely positive maps.
t
a For a countable group G there is also a notion of conditionally negative definite
m
functions ψ : G → C which satisfy ψ(g−1) = ψ(g) and the condition: ∀n ∈ N,
:
v α ,α ,...,α ∈ C, g ,g ,...,g ∈ G, if Σn α = 0 then Σn α α ψ(g−1g ) ≤ 0.
i 1 2 n 1 2 n i=1 i i,j=1 j i j i
X Real valued conditionally negative definite functions can be viewed as cocycles b ∈
r B1(G,π) where π : G → O(H) is an orthogonal representation of G (see [BdHV]).
a
Real valued conditionally negative definite functions can also be viewed as generators
of semigroups of positive definite functions by Schoenberg’s Theorem. These equiva-
lences then make it possible for certain connections between 1-cohomology, condition-
ally negative definite functions, and positive definite deformations, for example the
Delorme-Guichardet Theorem [De] [G] which states that a group has property (T) of
2000 Mathematics Subject Classification. 46L10, 46L57, 22D25.
Key words and phrases. property (T), 1-cohomology, von Neumann algebras.
Typeset by AMS-TEX
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2 JESSE PETERSON
Kazhdan [Ka] if and only if the first cohomology vanishes for any unitary representa-
tion.
It was Evans who introduced the notion of bounded conditionally completely posi-
tive/negative maps [E] related to the study the infinitesimal generators of norm con-
tinuous semigroups of completely positive maps. He noted that this definition gives
an analogue to conditionally positive/negative definite functions on groups. We will
extend the notion of conditionally completely negative maps to unbounded maps and
use a GNS type construction to alternately view them as closable derivations into a
Hilbert N-N bimodule. This is done in the same spirit as ([S1],[S2]) where Sauvageot
makesa connectionbetweenquantumDirichletforms, anddifferentialcalculus. Indeed,
it is shown in 1.7.1 that conditionally completely negative maps are in fact extensions
of generators associated to completely Dirichlet forms, however we are coming from a
different perspective here and so we will present the correspondence between condition-
ally completely negative maps and closable derivations in a way more closely related
to group theory.
In studying various properties of groups such as property (T) or the Haagerup
property one can give a characterization of these properties in terms of boundedness
conditions on conditionally negative definite functions (e.g. [AW]), hence one would
hope that this is possible for von Neumann algebras as well.
We will show that one can indeed obtain a characterization of property (T) in this
way. The main result is that a separable finite factor has property (T) if and only if
the 1-cohomology spaces of closable derivations vanish whenever the domain contains
a non-Γ set (see sec. 3 for the definition of a non-Γ set).
0.1. Theorem. Suppose that N is a separable finite factor, then the following condi-
tions are equivalent:
(1) N has property (T).
(2) N does not have property Γ and given any weakly dense ∗-subalgebra N ⊂ N,
0
1 ∈ N such that N contains a non-Γ set, we have that every densely defined closable
0 0
derivation on N into a Hilbert N-N bimodule is inner.
0
(3) There exists a weakly dense ∗-subalgebra N ⊂ N such that N is countably gener-
0 0
ated as a vector space and every closable derivation into a Hilbert N-N bimodule whose
domain contains N is inner.
0
This is the analogue to the Delorme-Guichardet Theorem for groups. As a corollary
we obtain that if X ,... ,X generate a finite factor with property (T), and if at least
1 n
one of the X ’s has diffuse spectrum then the derivations ∂ from [V1] cannot all be
j Xi
closable and hence the conjugate variables cannot all exist in L2(N,τ).
0.2. Corollary. Suppose that N is a finite factor with property (T), let X ,... ,X
1 n
generate N as a von Neumann algebra such that C[X ,... ,X ] contains a non-Γ set.
1 n
If one of the X ’s has diffuse spectrum then Φ∗(X ,... ,X ) = ∞.
j 1 n
A 1-COHOMOLOGY CHARACTERIZATION OF PROPERTY (T) 3
We also give an application showing that many amalgamated free products of finite
von Neumann algebras do not have property (T).
0.3. Theorem. Let N and N be finite von Neumann algebras with with normal
1 2
faithful tracial states τ and τ respectively, suppose that B is a common von Neumann
1 2
subalgebra such that τ | = τ | . If there are unitaries u ∈ U(N ) such that E (u ) =
1 B 2 B i i B i
0, i = 1,2. Then M = N ∗ N does not have property (T).
1 B 2
Otherthantheintroductionthereare4sections. Section1establishesthedefinitions
andnotationsaswellasgivestheconnectionbetween closablederivations, conditionally
completely negative maps, and semigroups of completely positivemaps. In section 2 we
characterize when a closable derivation is inner in terms of the conditionallycompletely
negative map and the semigroup. In Section 3 we state and prove the main theorem
(3.2), and in section 4 we give the application with amalgamated free products (4.2).
1. A GNS-type construction.
1.1. Conditionally completely negative maps. Let N be a finite von Neumann
algebra with normal faithful trace τ.
Definition. Suppose Ψ : N → L1(N,τ) is a ∗-preserving linear map whose domain
is a weakly dense ∗-subalgebra D of N such that 1 ∈ D , then Ψ is a conditionally
Ψ Ψ
completely negative (c.c.n.) map on N if the following condition is satisfied:
(1.1.1). ∀n ∈ N, x ,y ∈ D , j ≤ n, if Σn x y = 0 then Σn y∗Ψ(x∗x )y ≤ 0.
j j Ψ j=1 j j i,j=1 j j i i
It is not hard to see that condition (2.1.1) can be replaced with the condition:
(1.1.1)′. ∀n ∈ N, x ,y ∈ D , j ≤ n, if Σn x y = 0 then Σn τ(Ψ(x∗x )y y∗) ≤ 0.
j j Ψ j=1 j j i,j=1 j i i j
If φ : N → N is a completely positive map and k ∈ N then Ψ(x) = k∗x+xk−φ(x)
gives a map which is c.c.n. and bounded. If δ : N → L2(N,τ) is a derivation then δ is
c.c.n. Also if Ψ is a c.c.n. map and α : N → N is a τ-preserving automorphism then
Ψ′ = α◦Ψ◦α−1 is another c.c.n. map.
One can check that if Ψ and Ψ are c.c.n. such that D ∩D is weakly dense in
1 2 Ψ1 Ψ2
N, and if s,t ≥ 0, then Ψ = sΨ +tΨ is c.c.n. Also if {Ψ } is a family of c.c.n. maps
1 2 t t
on the same domain and Ψ is the pointwise k·k -limit of {Ψ } then Ψ is c.c.n.
1 t t
We say that Ψ is symmetric if τ(Ψ(x)y) = τ(xΨ(y)),∀x,y ∈ D . We say that Ψ is
Ψ
conservative if τ ◦ Ψ = 0. We also say that Ψ is closable if the quadratic form q on
L2(N,τ) given by D(q) = D , q(x) = τ(Ψ(x)x∗) is closable. Note that we will see in
Ψ
(1.3) that if Ψ : D → L2(N,τ) ⊂ L1(N,τ) is a conservative symmetric c.c.n. map
Ψ
then Ψ is automatically closable.
Note that if Ψ is a conservative symmetric c.c.n. map then τ(Ψ(1)x) = τ(Ψ(x)) =
0,∀x ∈ D , hence Ψ(1) = 0. Also note that if Ψ is symmetric and Ψ(1) ≥ 0 then given
Ψ
any x ∈ D , if we let x = x,x = 1,y = −1,y = x, then the above condition implies
Ψ 1 2 1 2
4 JESSE PETERSON
that τ(Ψ(x)x∗) ≥ 0, so that we actually have positivity instead of just the symmetry
condition.
1.2. Closable derivations. Let H be a Hilbert N-N bimodule, a derivation of N is a
(possibly unbounded) map δ : N → H which is defined on a weakly dense ∗-subalgebra
D of N such that 1 ∈ D , and such that ∀x,y ∈ D , δ(xy) = xδ(y) + δ(x)y. δ is
δ δ δ
closable if it is closable as an operator from L2(N,τ) to H.
δ is inner if δ(x) = xξ −ξx for some ξ ∈ H. δ is spanning if spD δ(D ) = H. δ is
δ δ
real if hxδ(y),δ(z)i = hδ(z∗),δ(y∗)x∗i , ∀x,y,z ∈ D .
H H δ
If δ′ : D → H′ is another derivation then we say that δ and δ′ are equivalent if
δ
there exists a unitary map U : H → H′ such that U(xδ(y)z) = xU(δ(y))z = xδ′(y)z
for all x,y,z ∈ D .
δ
Recall that if H is a Hilbert N-N bimodule then we can define the adjoint bimodule
Ho where Ho isthe conjugate Hilbertspace ofH and thebimodule structure is givenby
xξoy = (y∗ξx∗)o. If δ : D → H is a closable derivation then we may define the adjoint
δ
derivation δo : D → Ho by setting δo(x) = δ(x∗)o, then δo is a closable derivation and
δ
furthermore the derivations 1(δ + δo), and 1(δ − δo) are real derivations from D to
2 2 δ
H⊕Ho.
1.3. From conditionally completely negative maps to closable derivations.
Let Ψ be a conservative symmetric c.c.n. map on N with domain D . We associate
Ψ
to Ψ a derivation in the following way (compare with [S1]):
let H = {Σn x ⊗ y ∈ D ⊗ D |Σn x y = 0}. Define a sesquilinear form on
0 i=1 i i Ψ Ψ i=1 i i
H by hΣn x′ ⊗ y′,Σm x ⊗ y i = −1Σn Σm τ(Ψ(x∗x′)y′y∗). The positivity of
0 i=1 i i j=1 j j Ψ 2 i=1 j=1 j i i j
h·,·i is equivalent to the c.c.n. condition on Ψ. Let H be the closure of H after we
Ψ 0
mod out by the kernel of h·,·i . If p = Σn x ⊗ y such that Σn x y = 0 then
Ψ k=1 k k k=1 k k
x 7→ −1Σn τ(x∗xx Ψ(y y∗)) is a positive normal functional on N with norm hp,pi .
2 i,j=1 j i i j Ψ
Similarly y 7→ −1Σn τ(Ψ(x∗x )y yy∗) is a positive normal functional on N with
2 i,j=1 j i i j
norm hp,pi . We also have left and right commuting actions of D on H given by
Ψ Ψ 0
xpy = x(Σn x ⊗y )y = Σn (xx )⊗(y y), and by the preceeding remarks we have
k=1 k k k=1 k k
hxp,xpi = hx∗xp,pi ≤ kx∗xkhp,pi = kxk2hp,pi and hpy,pyi ≤ kyk2hp,pi for
Ψ Ψ Ψ Ψ Ψ Ψ
all x,y ∈ D . Hence the above actions of D pass to commuting left and right actions
Ψ Ψ
on H, and they extend to left and right actions of N on H given by the formulas:
hx[Σn x′ ⊗y′],[Σm x ⊗y ]i = Σn Σm τ(x∗xx′Ψ(y′y∗)),
i=1 i i j=1 j j H i=1 j=1 j i i j
h[Σn x′ ⊗y′]y,[Σm x ⊗y ]i = Σn Σm τ(Ψ(x∗x′)y′yy∗).
i=1 i i j=1 j j H i=1 j=1 j i i j
Since the above forms are normal the left and right actions commute and are normal
thus making H into a Hilbert N-N bimodule.
Define δ : D → H to be given by δ (x) = [x⊗1−1⊗x]. Then δ is a derivation
Ψ Ψ Ψ Ψ
such that
hδ (x),δ (y)i = hx⊗1−1⊗x,y ⊗1−1⊗yi
Ψ Ψ H Ψ
A 1-COHOMOLOGY CHARACTERIZATION OF PROPERTY (T) 5
1 1 1 1
= − τ(Ψ(y∗x))+ τ(Ψ(x)y∗)+ τ(Ψ(y∗)x)− τ(Ψ(1)xy∗)
2 2 2 2
= τ(Ψ(x)y∗),
for all x,y ∈ D . Also δ is real since
Ψ Ψ
hxδ (y),δ (z)i = hxy ⊗1−x⊗y,z ⊗1−1⊗zi
Ψ Ψ H Ψ
1 1 1 1
= − τ(Ψ(z∗xy))+ τ(Ψ(xy)z∗)+ τ(Ψ(z∗x)y)− τ(Ψ(x)yz∗)
2 2 2 2
1 1 1 1
= − τ(Ψ(1)z∗xy)+ τ(Ψ(z∗)xy)+ τ(Ψ(y)z∗x)− τ(Ψ(yz∗)x)
2 2 2 2
= h1⊗z∗ −z∗ ⊗1,1⊗y∗x∗ −y∗ ⊗x∗i = hδ (z∗),δ (y∗)x∗i ,
Ψ Ψ Ψ H
for all x,y,z ∈ D . If Ψ is closable then it follows that δ is closable. Also note that if
Ψ Ψ
Ψ : D → L2(N,τ) ⊂ L1(N,τ) then we would have that D = D(δ∗δ ) which would
Ψ Ψ Ψ Ψ
show that δ (and hence also Ψ) is closable.
Note that we will assume in addition that δ is spanning by restricting ourselves to
Ψ
spD δ(D ) ⊂ H.
Ψ Ψ
Also note that the requirement that Ψ(1) = 0 is not really much of a restriction
since if Ψ is any symmetric c.c.n. map with Ψ(1) ∈ L2(N,τ) then Ψ′(x) = Ψ(x) −
1Ψ(1)x− 1xΨ(1) defines a symmetric c.c.n. map with Ψ′(1) = 0.
2 2
1.4. From closable derivations to conditionally completely negative maps.
Let H be a Hilbert N-N bimodule and suppose that δ : N → H is a closable real
derivation defined on a weakly dense ∗-subalgebra D of N with 1 ∈ D .
δ δ
Let D = {x ∈ D(δ)∩N|y 7→ hδ(x),δ(y∗)i gives a normal linear functional on N}
Ψ
then by [S2] and [DL] D(δ)∩N is a ∗-subalgebra and hence one can show that D is
Ψ
a ∗-subalgebra of N. We define the map Ψ : D → L1(N,τ) by letting Ψ (x) be the
δ Ψ δ
Radon-Nikodym derivative of the normal linear functional y 7→ hδ(x),δ(y∗)i. Since δ
is closable Ψ is also closable.
δ
As δ is real Ψ is a symmetric ∗-preserving map such that τ ◦Ψ = 0 and if n ∈ N,
δ
x ,x ,...,x ,y ,y ,...,y ∈ D , such that Σn x y = 0 then:
1 2 n 1 x n Ψ i=1 i i
Σn τ(Ψ(x∗x )y y∗) = Σn hδ(x∗x ),δ(y y∗)i
i,j=1 j i i j i,j=1 j i j i H
= Σn hx∗δ(x ),y δ(y∗)+δ(y )y∗i +hδ(x∗)x ,y δ(y∗)+δ(y )y∗i
i,j=1 j i j i j i H j i j i j i H
= Σn hδ(x )y ,x δ(y )i +hx δ(y ),δ(x )y i
i,j=1 i i j j H i i j j H
= −2kΣn δ(x )y k2 ≤ 0.
i=1 i i H
Hence Ψ is a conservative symmetric c.c.n. map on D .
δ Ψ
6 JESSE PETERSON
Note that if we restrict ourselves to closable derivations which are spanning then an
easy calculation shows that the constructions above are inverses of each other in the
∼
sense that Ψ | = Ψ and δ = δ.
δΨ DΨ Ψδ
1.5. Closable derivations and c.c.n. maps from groups. Let Γ be a discrete
group, (C,τ ) a finite von Neumann algebra with a normal faithful trace, and σ a
0
cocycle action of Γ on (C,τ ) by τ -preserving automorphisms. Denote by N = C × Γ
0 0 σ
thecorrespondingcross-product algebrawithtraceτ givenbyτ(Σc u ) = τ (c ), where
g g 0 e
c ∈ C and {u } ⊂ N denote the canonical unitaries implementing the action σ on C.
g g g
Let (π ,H ) be a unitary or orthogonal representation of Γ, and let b : Γ → H be
0 0 0
an (additive) cocycle of Γ, i.e. b(gh) = π (g)b(h)+b(g), ∀g,h ∈ Γ. Set H to be the
0 π0
Hilbert space H0⊗RL2(N,τ) if π0 is an orthogonal representation and H0⊗CL2(N,τ) if
π is a unitary representation. We let N act on the right of H by (ξ⊗xˆ)y = ξ⊗(xˆy),
0 π0
x,y ∈ N,ξ ∈ H and on the left by c(ξ ⊗xˆ) = ξ ⊗(cˆx),u (ξ ⊗xˆ) = (π (g)ξ)⊗(uˆx),
0 g 0 g
c ∈ C,x ∈ N,g ∈ Γ,ξ ∈ H . Let D be the ∗-subalgebra generated by C and {u } ,
0 Γ g g
we define δ by δ (c u ) = c δ (u ) = b(g)⊗c ˆu , c ∈ C,g ∈ Γ, then we can extend
b b g g g b g g g g
δ linearly so that δ is a derivation on D . If (π ,H ) is an orthogonal representation
b b Γ 0 0
and 1 denotes the Dirac delta function at g then:
g
hcu δ (u ),δ (u )i = hπ (g)b(h),b(k)ihcuˆu ,uˆ i
g b h b k 0 g h k
= h−π (g)π (h)b(h−1),−π (k)b(k−1)ihcuˆu ,uˆ i1 (gh)
0 0 0 g h k k
= hb(k−1),b(h−1)ihuˆ∗,u∗uˆ∗c∗i
k h g
= hδ (u∗),δ (u∗)u∗c∗i,
b k b h g
for all g,h,k ∈ Γ, c ∈ C, thus showing that δ is real.
b
Also we have:
ˆ
|hδ (c u ),δ (Σ d u )i| = |Σ hb(g),b(h)ihcˆu ,d u i|
b g g b h∈Γ h h h∈Γ g g h h
= kb(g)k2|hc ˆu ,Σ d ˆu i| ≤ kb(g)k2kc kkΣ d u k ,
g g h∈Γ h h g h∈Γ h h 1
for all g ∈ Γ, c ∈ C, Σ d u ∈ D . Hence if x = Σ c u ∈ D , and y ∈ D then
g h∈Γ h h Γ g∈Γ g g Γ Γ
|hδ (x),δ (y)i| ≤ (Σ kb(g)k2kc k)kyk . In particular this shows that δ is closable.
b b g∈Γ g 1 b
Now suppose that ψ : Γ → C is a real valued conditionally negative definite function
on Γ such that ψ(e) = 0 and let (π ,b ) be the representation and cocycle which
ψ π
correspond to ψ through the GNS construction [BdHV]. Let (H,δ) denote the Hilbert
N-N bimodule and closable derivation constructed out of (π ,b ) as above and let Ψ
ψ π
be the symmetric c.c.n. map associated to (H,δ) as in 1.4. Then a calculation shows
that Ψ(Σ c u ) = Σ ψ(g)c u , and in fact it is an easy exercise to show that even if
g g g g g g
ψ is not real valued Ψ(Σ c u ) = Σ ψ(g)c u still describes a c.c.n. map.
g g g g g g
A 1-COHOMOLOGY CHARACTERIZATION OF PROPERTY (T) 7
Conversely, if (H,δ) is a Hilbert N-N bimodule and a closable derivation such that
δ is defined on the ∗-subalgebra generated by C and {u } then we can associate to it
g g
a representation π on H = sp{δ(u )u∗|g ∈ Γ} by π (g)ξ′ = u ξ′u∗, ξ′ ∈ H . Also we
0 0 g g 0 g g 0
may associate to δ a group cocycle b on Γ by b(g) = δ(u )u∗, g ∈ Γ. If Ψ is a c.c.n.
g g
map which is also defined on the ∗-subalgebra generated by C and {u } then we
g g
can associate to it a conditionally negative definite function ψ by ψ(g) = τ(Ψ(u )u∗).
g g
Furthermore if δ is real then by taking only the real span above we have that H is a
0
real Hilbert space and π is an orthogonal representation, also ψ is real valued if and
0
only if Ψ is symmetric, and if (H,δ) and Ψ correspond to each other as in 1.3 and 1.4
then (π ,b) and ψ correspond to each other via the GNS construction.
0
1.6. Examples from free probability. From above we have two main examples
of closable derivations, those which are inner, and those which come from cocycles
on groups. In [V1] and [V2] Voiculescu uses certain derivations in a key role for his
non-microstates approach to free entropy and mutual free information. We will recall
these derivations which will give us more examples of closable derivations under certain
circumstances.
1.6.1 The derivation ∂ from [V1]. Suppose B ⊂ N is a ∗-subalgebra with 1 ∈ B and
X
X = X∗ ∈ N. If we denote by B[X] the subalgebra generated by B and X, and if X
and B are algebraically free (i.e. they do not satisfy any nontrivial algebraic relations)
then there is a well-defined unique derivation
∂ : B[X] → B[X]⊗B[X] ⊂ L2(N,τ)⊗L2(N,τ)
X
such that ∂ (X) = 1⊗1 and ∂ (b) = 0 ∀b ∈ B.
X X
We note that if ∂ is inner then by identifying L2(N,τ)⊗L2(N,τ) with the Hilbert-
X
Schmidt operators we would have that there exists a Hilbert-Schmidt operator which
commutes with B. Therefore if B contains a diffuse element (i.e. an element which
generates a von Neumann algebra without minimal projections) then we must have
that ∂ is not inner.
X
Recall from [V1] that the conjugate variable J(X : B) of X w.r.t. B is an element
in L1(W∗(B[X]),τ) such that τ(J(X : B)m) = τ ⊗τ(∂ (m)) ∀m ∈ B[X], i.e. J(X :
X
B) = ∂∗ (1⊗1).
X
If J(X : B) exists and is in L2(N,τ) (as in the case when we perturb a set of
generators by free semicircular elements) then by Corollary 4.2 in [V1] we have that
∂ is a closable derivation.
X
1.6.2 The derivation δ from [V2]. Suppose A,B ⊂ N are two ∗-subalgebras with
A:B
1 ∈ A,B. If we denote by A∨B the subalgebra generated by A and B, and if A and
B are algebraically free then we may define a unique derivation
δ : A∨B → (A∨B)⊗(A∨B) ⊂ L2(N,τ)⊗L2(N,τ)
A:B
8 JESSE PETERSON
by δ (a) = a⊗1−1⊗a ∀a ∈ A, and δ (b) = 0 ∀b ∈ B.
A:B A:B
We note that for the same reason as above if B contains a diffuse element and A 6= C
then we must have that the derivation is not inner.
Recall from [V2] that the liberation gradient j(A : B) of (A,B) is an element
in L1(W∗(A ∪ B),τ) such that τ(j(A : B)m) = τ ⊗ τ(δ (m)) ∀m ∈ A ∨ B, i.e.
A:B
j(A : B) = δ∗ (1⊗1).
A:B
If j(A : B) exists and is in L2(N,τ) then by Corollary 6.3 in [V2] δ is a closable
A:B
derivation.
1.7. Generators of completely positive semigroups. Let N be a finite von
Neumann algebra with normal faithful trace τ. A weak*-continuous semigroup {φ }
t t≥0
on N is said to be symmetric if τ(xφ (y)) = τ(φ (x)y), ∀x,y ∈ N, and completely
t t
Markovian if each φ is a unital c.p. map on N. We denote by ∆ the generator of a
t
symmetriccompletely Markoviansemigroup {φ } onN, i.e. ∆isthedensely defined
t t≥0
operator on N described by D(∆) = {x ∈ N : x−φt(x) has a weak limit as t → 0},
t
and ∆(x) = lim x−φt(x), we also denote by ∆ the generator of the corresponding
t→0 t
semigroup onL2(N,τ). Then ∆describes a completely Dirichlet form [DL] on L2(N,τ)
by D(E) = D(∆1/2), E(x) = k∆1/2(x)k2.
2
From [DL] we have that D(E)∩N is a weakly dense ∗-subalgebra and hence it fol-
lows from [S1] that there exists a Hilbert N-N bimodule H and a closeable derivation
δ : D(E) ∩N → H such that E(x) = kδ(x)k2, ∀x ∈ D(E) ∩ N. Conversely it follows
from [S2] that if D(δ) is a weakly dense ∗-subalgebra with 1 ∈ D(δ) and δ : D(δ) → H
is a closable derivation, then the closure of the quadratic form given by kδ(x)k2 is com-
pletely Dirichlet on L2(N,τ) and hence generates a symmetric completely Markovian
semigroup as above (see also [CiSa]).
From sections 1.3 and 1.4, and from the remarks above we obtain the following.
1.7.1. Theorem. Let N ⊂ N be a weakly dense ∗-subalgebra with 1 ∈ N and
0 0
suppose Ψ : N → L1(N,τ) is a closable, conservative, symmetric c.c.n. map, such
0
that Ψ−1(L2(N,τ)) is weakly dense in N. Then ∆ = Ψ|Ψ−1(L2(N,τ)) is closable as a
densely defined operator on L2(N,τ) and ∆ is the generator of a symmetric completely
Markovian semigroup on N. Conversely if ∆ is the generator of a symmetric completely
Markovian semigroup on N then ∆ extends to a conservative, symmetric c.c.n. map
Ψ : N → L1(N,τ) where N is the ∗-subalgebra generated by D(∆).
0 0
2. A characterization of inner derivations.
Let N be a finite von Neumann algebra with normal faithful trace τ. Given a
symmetric c.c.n. map Ψ on N we will now give a characterization of when Ψ is norm
bounded.
A 1-COHOMOLOGY CHARACTERIZATION OF PROPERTY (T) 9
2.1. Theorem. Let Ψ : D → L1(N,τ) be a closable, conservative, symmetric c.c.n.
Ψ
map with weakly dense domain D . Let δ : D → H be the closable derivation associ-
Ψ Ψ
ated with Ψ. Then the following conditions are equivalent:
(a) δ extends to an everywhere defined derivation δ′ which is inner and such that given
any x ∈ N there exists a constant C > 0 such that |hδ′(x),δ′(y)i| ≤ C kyk , ∀y ∈ N.
x x 1
(b) There exists a constant C > 0 such that |hδ(x),δ(y)i|≤ Ckxkkyk , ∀x,y ∈ D .
1 Ψ
(c) Ψ is norm bounded on (D ) .
Ψ 1
(d) The image of Ψ is contained in N ⊂ L1(N,τ) and −Ψ extends to a mapping which
generates a norm continuous semigroup of normal c.p. maps.
(e) There exists k ∈ N and a normal c.p. map φ : N → N such that Ψ(x) = k∗x +
xk −φ(x), ∀x ∈ D .
Ψ
Proof. (a) ⇒ (c): Let δ′ be the everywhere defined extension of δ, and let Ψ′ be the
c.c.n. map associated with δ′. Since given any x ∈ N there exists a constant C > 0
x
such that |hδ′(x),δ′(y)i| ≤ C kyk , ∀y ∈ N we have that the image of Ψ′ is contained in
x 1
N. Also since Ψ′(1) = 0 we have that for all x ∈ DΨ′, Ψ′(x∗x)−x∗Ψ′(x)−Ψ′(x∗)x ≤ 0
and so −Ψ′ is a dissipation ([L],[Ki]). As −Ψ′ is also everywhere defined, it is bounded
by Theorem 1 in [Ki].
(b) ⇔ (c): Suppose (b) holds, then for all x,y ∈ D ,
Ψ
|τ(Ψ(x)y∗)| = |hδ(x),δ(y)i| ≤ Ckxkkyk .
1
So by taking the supremum over all y ∈ D such that kyk ≤ 1 we have that kΨ(x)k ≤
Ψ 1
Ckxk, ∀x ∈ D .
Ψ
Suppose now that Ψ is bounded by C > 0. Then for all x,y ∈ D ,
Ψ
|hδ(x),δ(y)i|= |τ(Ψ(x)y∗)| ≤ kΨ(x)kkyk ≤ Ckxkkyk .
1 1
(c) ⇒ (d): This follows from [E] Proposition 2.10.
(d) ⇒ (e): This is Theorem 3.1 in [ChE].
(e) ⇒ (a): Suppose that for k ∈ N and φ c.p. we have Ψ(x) = k∗x+xk−φ(x), ∀x ∈ N.
Let φ′ = τ(φ(1))−1φ and let (H,ξ) be the pointed Hilbert N-bimodule associated with
φ′. Hence if we set δ′(x) = (τ(φ(1))/2)1/2[x,ξ] then we have δ′ ∼= δ. By replacing
k with 1(k + k∗) and φ with 1(φ + φ∗) we may assume that φ is symmetric, it is
2 2
then easy to verify that there exists a constant C > 0 such that for all x,y ∈ N,
|hδ′(x),δ′(y)i| ≤ Ckxkkyk . Hence δ′ gives an everywhere defined extension of δ which
1
satisfies the required properties. (cid:3)
Our next result is in the same spirit as Theorem 2.1 and provides several equivalent
conditions for when a closable derivation is inner.
10 JESSE PETERSON
2.2. Theorem. Let Ψ : D → L1(N,τ) be a closable, conservative, symmetric c.c.n.
Ψ
map with weakly dense domain D . Let δ : D → H be the closable derivation associ-
Ψ Ψ
ated with Ψ. Then the following conditions are equivalent:
(α) δ is inner.
(β) δ is bounded on (D ) .
Ψ 1
(γ) Ψ is k·k -bounded on (D ) .
1 Ψ 1
(δ) Ψ can be approximated uniformly by c.p. maps in the following sense: for all ε > 0,
there exists k ∈ N, and φ a normal c.p. map such that kΨ(x)−k∗x−xk+φ(x)k ≤ εkxk,
1
∀x ∈ D .
Ψ
Proof. (α) ⇒ (δ): Suppose ξ ∈ H such that δ(x) = xξ − ξx, ∀x ∈ D . Let ε > 0.
Ψ
Since the subspace of “left and right bounded” vectors is dense in H, let ξ ∈ H such
0
that there exists a constant C > 0 such that kxξ k ≤ Ckxk , ∀x ∈ N, kξ k ≤ kξk, and
0 2 0
also kξ − ξ k < ε/8kξk. As ξ is “bounded” we may let φ be the normal c.p. map
0 0 ξ0
associated with ξ /kξ k. Let φ = 2kξ k2φ , and let k = φ(1)/2.
0 0 0 ξ0
Note that since δ is real we have that ξ is also real, i.e. hxξ ,ξ yi = hy∗ξ ,ξ x∗i,
0 0 0 0 0
∀x,y ∈ N.
Then if x,y ∈ D we have:
Ψ
τ((Ψ(x)−k∗x−xk +φ(x))y∗)
1 1
= τ(Ψ(x)y∗)− τ(φ(1)xy∗)− τ(xφ(1)y∗)+τ(φ(x)y∗)
2 2
= hδ(x),δ(y)i−hxy∗ξ ,ξ i−hy∗xξ ,ξ i+2hxξ y∗,ξ i
0 0 0 0 0 0
= hxξ −ξx,yξ−ξyi−hxξ −ξ x,yξ −ξ yi.
0 0 0 0
Hence:
|hΨ(x)−k∗x−xk +φ(x),yi|
≤ kxξ −ξxkkyξ−ξy−yξ +ξ yk+kyξ −ξ ykkxξ −ξx−xξ +ξ xk
0 0 0 0 0 0
≤ 4kxkkξkkykkξ−ξ k+4kykkξ kkxkkξ −ξ k
0 0 0
≤ εkxkkyk.
Thus by taking the supremum over all y ∈ (D ) we have the desired result.
Ψ 1
(δ) ⇒ (γ): Let k ∈ N and φ c.p. such that kΨ(x)−k∗x−xk+φ(x)k ≤ kxk. By ([P2]
1
1.1.2) kφ(x)k ≤ kφ(1)k kxk, ∀x ∈ N. Hence for all x ∈ D :
2 2 Ψ
kΨ(x)k ≤ kΨ(x)−k∗x−xk +φ(x)k +kk∗x−xk +φ(x)k
1 1 2
≤ (1+2kkk +kφ(1)k )kxk.
2 2