Commun.
math. Phys. 13, 142—153 (1969)
On
Some Groups of Automorphisms
of von
Neumann
Algebras
with
Cyclic and Separating Vector
A. Z. JADCZYK
Institute
of Theoretical Physics, University of Wroclaw
Received January 6, 1969
Abstract. Let
$1
be a von Neumann algebra with the vector Ω cyclic and separa-
ting for $1. Let ^β
Θ
be a group of unitary operators under which both Ω and 21
are
invariant. Let 33 (resp. 9Γ) be the
fixed
point algebra in 21 (resp. in 2Γ). Let F
o
be an orthogonal projection onto the subspace of all vectors invariant under 93
G
.
Tt
is shown that 9ί = (21 \J
{F
o
})"
and that the irreducibility of 91 implies that F
o
is one-dimentional. Other consequences of the Theorem of
KOVACS
and Sziics are
also derived. In sec. 3. the spectrum properties of the group %}
G
are studied. It is
proved that the point spectrum of 93
G
is symmetric and that it is a group provided
91 is irreducible. In this case there
exists
a homomorphism χ -» χ (resp. χ -> χ)
of the point spectrum of 93
C
into the group of unitary operators in 21 (resp. in 2Γ)
uniquely (up to the phase) defined by χ V
g
= χ(g) V
g
χ (resp. the same for χ).
In
sec. 4. the application of the foregoing results to the KMS-Algebra is given.
1.
Introduction
It
has been shown in [1] that the state of thermal equilibrium of an
infinite system is mathematically described by the state ω (over the 0*-
algebra of
observables
21^)
satisfying
the KMS boundary conditions. Let
(π, Jf
π
) be a canonical representation defined by ω, and let Ω be the
vector representing the state ω in ^f
π
. Finally, let 21 = π(Qί^)
f/
. It has
been exhibited in [1] that there is a peculiar symmetry between 2ί and 2Γ.
In
particular, the vector Ω invariant under time and space translations
is cyclic for both, 21 and 2Γ. A similar situation
arises
when one deals
with internal symmetries in the framework of
Algebraic
Quantum Field
Theory. The goal of our paper is to study a general situation, when there
is
given
a von Neumann algebra 21, the group
Λ-^VgΛV^
1
of auto-
morphisms of 21 and the vector Ω invariant under all V
g
and cyclic for
21 and 2Γ. A general discussion of such a situation is
given
in Section 2.
Our
tool, in this section, is the theorem of Kovics and Sziics (see [4] and
[4a, Theorem 1]). It is shown that the irreducibility of 21 w {V
g
} implies
that
Ω is the unique vector invariant under all V
g
. We also show that
(21 \J
{Vg})"
= (21 \J
{FQ})",
where F
o
is a projection onto the subspace
Groups
of Automorphisms 143
of all vectors invariant under { V
g
}. In Section 3 we deal with the spectrum
properties of the group 93#. The results of this section are closely related
to
those obtained by D. W.
ROBINSON
[3]. In particular we prove that
the
point spectrum of 93# is symmetric and it is a group provided 93^
commutes
with another group 93^ such that 93^ w 21 is irreducible. The
Theorem
3.5. is the main result of this section. Finally, in Section 4. we
apply the results of the foregoing sections to the case of the KMS-Algebra.
In
particular, for each discrete eigenvalue of the "momentum" p we
construct
two unitary operators p ζ 21 and p ζ 2Γ such that
It
is shown that for each such p, the vector ψ
P
= pΩ also describes the
state of thermal equilibrium, with the same temperature as Ω.
2.
General
Discussion
In
this section ψe are concerned with some properties of groups of
automorphisms of a von
Neumann
algebra with a cyclic and separating
vector. The following results are a simple application of the theorem of
Kovics and Szϋcs (see [4] and (4a, Theorem 1]).
2.1.
Notation.
Let 21 be a von Neumann
algebra
acting
on a
Hilbert
space
J4f,
with
commutant 21 and the
center
3 Let G be a
group
and g -> V
g
be a unitary
representation
of G on ^
such
that A{g) =
VgAVj
1
is in 21
for
each
A in 21. Let F
o
be a
projection
onto
the
subspace
of all ψ in
£#*,
invariant under all V
g
.
Denoting
by
93
#
an
image
of G under the
mapping
g -^
Vglet^Kbe
a von Neumann
algebra
generated
by 21 and ^g, and lei 35
be a
fixed
point
algebra
in 21 i.e. 33 = 21 r\ 93^. We remark that 21' is now
also
invariant under θ,
with
a
fixed
point
algebra
91'.
Finally,
let ~£ be a
Godement
mean
over
G {see e.g. [4a]). This
notation
will
be
fixed
throughout
this
section.
2.2. Lemma.
Assume
there
exists
a unit
vector
Ω
which
is
cyclic
and
separating
for 2t and
such
that V
g
Ω = Ω for all g in G. Then
I. There
exist
two unique
normal
G-invariant
projection
maps
Φ and Φ'
from
21
onto
93 and
from
21'
onto
91'
respectively.
Φ and Φ' are
positive
and
faithful.
Φ(A)
(resp.
Φ' (A)) can be
defined
(equivalently) as
i)
the unique element of 21 r\ conv{^L ($)}-
(resp.
21' r\ conv {.4
($)}-)
ii) the unique element of 2ί (reap. 21')
such
that Φ(A)F
Q
=
F
0
AF
Q
(resp.Φ'(A)F
0
= F
0
AF
0
),
in)
the unique
operator
on Jf satisfying(φ, Φ(A)ψ) = ~# {(φ, A(g)ψ)}
for all φ,ψ ζJ4?
(resp.
the
same
for Φ
f
(A)),
II.
A
normal
linear
from
ω on<Ά
(resp.
2Γ) is
G-invariant
if and
only
if ω = (α>|93) o Φ
(resp.
ω = (ω|9T) o Φ').
144 A. Z.
JADCZYK:
///.
With
A in
21
{resp.
21')
the
foϊloiving
are
equivalent
(p)
F
0
A = AF
0
,
(pp)
A ζ
93 (resp.
4 ξ
91')
IF.
Proof.
I,
i)—iϋ)
are an
immediate consequence
of [4a,
Theorem
1].
To
prove
III
suppose
F
0
A =
^4i^
0
where
^4 is in
21. Then
AF
Q
=
F
0
AF
0
and
we
find that,
by
ii),
A =
Φ(^4)
ζ
95. The case
of
^4
ζ
2Γ
is
handled
in
exactly the same
way.
Implication (pp) -> (p)
is
obvious. Finally
we
will
prove
IV. It is
clear that
JP
0
^
[93 Ω]
so it is
sufficient
to
check that 93 Ω
is dense
in
F
0
J4f.
Given
ψ ζ F
0
J^ we can
choose
a
sequence
A
n
£ 21 such
that
A
n
Ω
->
ψ. But
then
V
- F
o
ψ =
lim^o^Ω
=
limΦ^Jβ
and
the proof
is
complete.
As
a
corollary
of the
foregoing lemma
we
have
2.3.
Corollary.
Let G be as in
Lemma 2.2. Then
i)
and 93
is
abelian
(resp.
finite,
semi-finite,
properly
infinite,
purely
infinite)
if
and
only
if 9l
r
is
abelian
(resp.
finite,
semi-finite,
properly
infinite,
purely
infinite).
Moreover,
if 93 is
abelian
then
93^
93-F
o
=
9t'.F
0
^
91'.
ii)
If G
x
is
another
group
satisfying
assumptions
of
Lemma
2.2
then
one
has an
equivalence
of
(P)
(ppp)
F
01
^ F
o
In
particular,
if for all A, B ξ 2ί and φ, ψ ζF
01
ffl
Jέ
{(φ, [
V
s
A Vj\
B]y,)}
= 0, gζG,
and
8
C 93, then
all (p)
—
(ppp) are
satisfied.
Proof.
The
first
statement
follows
by [4a,
Corollary
2]. Now, by
[7,
Chapter III, §
2,
Prop. 3],
if
93
is
abelian etc.
so
does 91'. On the other
hand,
if 93 is
abelian, then 93i^
0
,
9^JP
0
an(
i 9?'
are
abelian and therefore
93.F
O
=
(91
Λ
9T).F
0
-
9T-F
0
. Implications
(^)
->
(^^) and (^)
->
(^p)
are
an
immediate consequence
of
Lemma
2.2, IV. On the
other hand,
if
A
is in 93
2
then F
01
AΩ
and if (ppp)
holds then also F
0
AΩ
= AΩ or
Φ(A)Ω
=
^4ί3. Thus
J. =
Φ(^4) £93. Implication (ppp) ->
(^) is ob-
tained
in a
much
the
same
way.
Finally,
if 93 ^ 8 and G
1
satisfies
the
last assumption
of the
Corollary, then
it
follows
from [4 a, Lemma
2]
that
&ί
C 3>
and so % C 8
C 9T. Q.E.D.
Groups
of
Automorphisms
145
Our
next task
is to
show that
the
cyclicity
of Ω for 2Γ
can
in a
sence
replace
the
asymptotic abelianness. This
can be
also seen
in the
next
section, where
the
spectrum properties
are
studied.
The
theorem given
below should
be
compared with
[5,
Theorem
6]. (A
similar statement,
stated
in a
less
general form has been proved
by
H.
ARAKI
and H.
MIYATA
[8])
2.4. Theorem. With
notation
and
assumptions
of
Lemma
2.2. the
following
are
equivalent.
i) The
state
ω(Λ) =
(Ω,
AΩ) is an
extremal
G-invariant
state
over
21,
ii)
9t' = {λl},
ϋi)
F
o
is
one-dimentional,
iv)
23 = {A/},
v)
Φ(A) =
(Ω,
AΩ) for all A in
31,
vi)
ω is
weakly
clustering
Jί
{ω(A(g)B)}
= ω{A) ω(B) for all A, B
£31,
vϋ)
ω
satisfies
the
"stability"
condition:
a)
J?{ω(B*A(g) B)} = ω(B*B)ω(A) for all A, B £21,
b)
Jί{ωicA{g)B)}
= ω{CB)ω(A) for all A, B, G $21.
Proof. For
implications
i)
<=>
ii)
«-iϋ)-»
iv) see for
example
[9],
Theorem
4.
Suppose
now
that
F
o
= [Ω].
Then,
by
Lemma
2.2, IV, we
have
AΩ = λΩ for A in 23
and
A in
2T. Hence
23 = {λl} and 9T = {λl}.
It
is
evident that
iv)
implies
v) and if v)
holds then
Jί
{ω(CA
(g)B)}
= Jί {(β, CA
(g)
BΩ)} = (Ω, CΦ(A) BΩ)
=
(Ω,AΩ)(Ω
}
CBΩ).
But this means that
it is
sufficient
to
prove
vi)
->
v).
However,
the
latter
is obvious
if we
notice that
Jί
{ω(A
(g)B)}
- (Ω, Φ(A) BΩ)
and
Ω
is.separating
for 21.
3.
Properties
of the
Spectrum
It
is of
some interest that some group properties
of the
spectrum,
typical
for
asymptotically abelian systems
(see e.g.
[10], Theorem
3a)
appear also
in the
situation discussed
in the
preceding section.
Our
method
of
exhibiting these properties makes
it
possible
to
adapt
the
considerations
of
this section
to the
study
of the
spectrum
of
internal
symmetries
in the
Algebraic Quantum Field Theory
(see
[11, lla]).
3.1.
Definition.
Let G
-> 23^
be a
unitary representation
of
the group
G
on
a
Hubert space
Jf. Let Q be the
group
of all
(bounded) characters
of
the
group
G.
With
χ in ύ we say
that
χ is in a
point spectrum
of
23#
if
there
is 0
=f=
ψ ζ 3tf
such that
V
g
ψ
= χ(g)ψ for all g £G . (*)
10 Commun.math.Phys.,
Vol. 13
146 A. Z. JADCZYK:
Let σ(93
G
) be the set of all such characters. Let for each χ ζ 6, F(χ) be
a
projection onto the (closed) subspace of all ψ in Jή?
satisfying
(*). It is
clear that χ ( σ(93<?) if and only if F(χ)
=j=
0, and by
(ψ,F(χ)ψ)
= Jί
{χ(g)-i
(φ, F- ψ)} ^ Jt {χ(g) (φ, V^y)}
if and only if the righ-hand side is non-zero for some φ, ψ ζ 3ff.
3.2. Theorem. Let 21 and {V
g
} be as in 2.1.
Assume
that F
o
has
central
carrier
I in 31 (or
equivalently,
is
cyclic
for 21), and
93
#
commutes
with
some
other
$}
Gi
such
that
93
Gi
2l95g
l
= 21 and 93^ r\ 21 = {λl}. Then σ(¥>
G
) is a
subgroup
of the
character
group
0 of G.
Proof.
We
first
observe that, due to the irreducibility of 93^ \j 2t
7
,
F(χ) is cyclic for 93^ w 21' for each χ in σ(93
G
). On the other
hsinάF(χ)J^
is an invariant subspace for 93^, and therefore must be cyclic for 21'
(we also have 93^2l
/
93*
i
= 21'). Now, let χ, χ
^a(^
G
).
For an arbitrary
φ,
ψ (: Je, A ζ 21, B ζ 21' 'we then have
^{χ{g)'KF{χ')
φ,A(g)
B* F
o
ψ)} = Jt
{χigYKF{χ')
φ,
B*V
S
AF
o
ψ)}
=
(BF(χ')φ
9
F(χ)AF
o
ψ).
On
the other hand
l
=
(F(χ')φ,AF(χ-iχ')B*F
o
ψ)
so we have
(BF(χ')φ
9
F(χ)AF
o
ψ)
=
(F(χ')φ,
AF(χ~^χ
f
)
B*F
o
ψ)
for all A ζ 2ί, ^ ζ 2C and 99, ^ in ^f
7
. However, by the hypothesis,
is cyclic for 21' and F
o
is cyclic for 21, so we conclude that the left-hand
side is non-zero for some A, B, φ, ψ and therefore F(χ~
1
χ) Φ 0 or
X~
λ
X ί ^(53c/) I*
follows
that with χ ζ σ(93
G
) also χ-
1
is in σ(93
G
) and if
χ
and ^' are in σ(^3
G
) then ^ χ' does also. Q.E.D.
By putting χ' = 1 in the last equality we see that F(χ~
1
) 4= 0 once #
0
is cyclic for 21 and 2Γ. So we have
Corollary.
With assumptions of Lemma 2.2. the point spectrum of 93#
is symmetric
σ(93
β
)
=
σ(93
β
)-
1
.
In
some cases it occurs that there is an involution J on Jf transforming
21 onto 2Γ. This also implies, as was already pointed out by M.
WINNINK
[2,
IV.5, lemma], the symmetry of the point spectrum of 93^. Let us
note
that the following theorem is generally true
3.3.
Theorem.
Let 93^ be a
group
of unitary
operators
acting
on a
Hiΐbert
space
Jf.
Assume
there
exists
an
involution
J
commuting
with
93^.
Then the
point
spectrum
of 93^ is
symmetric
and
JF(χ)J
= F(χ~
1
).
Moreover,
if 93# is abelian,
n-parameter
and
strongly
continuous
then
JF(Δ)J = F{-Δ) for
each
Borel
set Δ C R
n
.