Commun.
math. Phys.
30,129—152
(1973)
© by
Springer-Verlag
1973
Local
Normality
in
Quantum Statistical Mechanics
Masamichi
Takesaki
Department
of
Mathematics, University
of
California,
Los
Angeles,
USA
Marinυs Winnink
Institute
for
Theoretical Physics, University
of
Groningen, Groningen,
The
Netherlands
Received
October
2,
1972
Abstract.
It is
shown that K.M.S.-states
are
locally normal
on a
great number
of
C*-algebras that
may be of
interest
in
Quantum Statistical Mechanics.
The
lattice structure
and the
Choquet-simplex structure
of
various sets
of
states
are
investigated.
In
this respect
special attention
is
payed
to the
interplay
of the
K.M.S.-automorphism group with other
automorphism
groups under whose action K.M.S.-states
are
possibly invariant.
A
seemingly weaker notion than
G-abelianness
of the
algebra
of
observables, namely
G'-
abelianness,
is
introduced
and
investigated. Finally
a
necessary
and
sufficient
condition
(on
a
C*-algebra with
a
sequential separable
factor
funnel)
for
decomposition
of a
locally
normal state into locally normal states
is
given.
§
1.
Introduction
The
investigation
of
representations
of the
commutation relations
and
anticommutation
relations
by
DelΓAntonio,
Doplicher
and
Ruelle
[3]
has
forced upon
us the
concept
of a
locally normal state.
Due to
this
origin, locally normal states have been studied
on a
C*-algebra
which
is a
C*-inductive limit
of
sub-C*-algebras, which
in
essence
are
irreducible
C*-algebras
on
suitably chosen Hubert-spaces
[14,21].
In
[12]
the
concept
of a
locally normal state
has
been generalized
for
a
C*-algebra that
is a
C*-inductive limit
of a net of von
Neumann
algebras.
In the
case where
the net
consists
of
factors each having
a
representation
on a
separable
Hubert
space,
the net is
called
a
funnel.
We
shall
use the
word
funnel
for the net of von
Neumann
(or
rather W*)-
algebras
that generate
a
C*-algebra
U
in the
sense
of
[12],
Definition
2,
even
if the net
does
not
contain only factors
and
even
if the net
contains
factors
which
are not of the
same type.
It
is the aim of
this
paper
to
investigate
the
locally normal character
of
K.M.S. states
on a
C*-algebra
U
with
a
funnel
whose components
are
σ-finite
properly
infinite
W
/
*-algebras.
As a
result
we
find
then that
every
K.M.S. state
on U is
locally normal.
It
then follows that
ω
is
normal
on
every
finite
factor contained
in
It
and on
every
σ-finite
properly
130
M.
Takesaki
and M.
Winnink:
infinite
W*-algebra
contained
in
U.
If we
further
specialize
the
funnel
that
generates
U,
to a
sequential
funnel
of
properly
infinite
W
/
*-algebras
with
a
separable
predual
1
,
then
the
cyclic representation induced
by a
K.M.S.
state acts
on a
separable Hubert-space
and
consequently every
K.M.S. state
is
normal
on
every
factor
contained
in U.
Also here
ω
is
normal
on
more general
VF*-algebras
than factors
in U,
namely
on
every
σ-finite
properly
infinite
W*
-algebra
contained
in U.
In
the
case
of a
sequential
funnel
of
properly
infinite
W*-algebras
with
a
separable predual
we
prove that every non-primary K.M.S. state
admits
a
unique decomposition into extremal K.M.S. states under
suitable
conditions
on the
K.M.S. automorphism, namely that
the set of
K.M.S. states
for a
given K.M.S. automorphism
at a
given temperature
is
compact
in the
weak*-topology
on U*. In the
course
of the
argument
we
show that every weak*-comρact subset
of the set of
locally normal
states
is
metrizable compact.
The
simplex structure
of
various sets
of
states
is
investigated.
For
example,
we
shall show that
the set of
K.M.S. states which
are
also
invariant under
the
action
of an
arbitrary group
of
automorphisms that
is
different
from
the
K.M.S. automorphism group
is a
simplex. Further-
more
we
shall investigate
to
what extent
a
locally normal state
can be
decomposed into locally normal states.
§
2.
Notation,
Definitions
and
Preliminary
Results
Let
9JΪ
be a
W*-algebra,
then
by
definition
3R
is the
Banach-space
dual
of a
Banach-space
SR*,
i.e.
9W
-
(2RJ*
9K*
is
called
the
predual
of
9JI
and
consists precisely
of
those elements
in
9K*
(the
Banach-space dual
of
9M)
which
are
σ(SDΐ,
9WJ
continuous.
Because
9JΪ
-
(9KJ*,
the
<j(9Jl,
9ϊy
topology
on
9W
is the
weak*-
topology
on
9W.
9W
is
said
to be
separable,
whenever
the
Banach-space
ΪR^
is
separable.
For
SOΐ
separable
the
unit ball
in
9W
is
metrizable
and
since
it
is
compact
in the
σ(9Jί,
501^)
topology
it is
separable;
we can
therefore
find a
separable C*-algebra
91
which
is
σ(9Jl,
9JΪJ
dense
in
$)ϊ.
Let
{50ί
α
}
αeΓ
be a net of
P^*-algebras
in the
sense
of
[12],
Definition
2,
i.e.
i)
to all
pairs
9Jζ,
$Jl
β
in
{9Jί
α
}
α6Γ
there exists
9ϊl
r
with
the
property
ii)
Every
S0ΐ
α
contains
the
unit
of U
where
U is
defined
in
iii)
U-
(j
9Jϊ
α
"
((")"
is the
norm closure
of
(•)).
αeΓ
A
C*-algebra
U has a
funnel
{SOΐJαgΓ?
whenever
{9W
α
}
α6Γ
satisfies
i),
ii) and
iii)
above.
9Jί
α
is
called
a
component
of the
funnel.
A
C*-algebra
1
For
definitions
see § 2.
Quantum
Statistical Mechanics
131
U
has a
σ-finite
(resp. separable) properly
infinite
funnel
whenever
it
has a
funnel
whose components
are
σ-finite
(respectively separable)
properly
infinite
VF*-algebras.
A
VF*-algebra
931
is
separable
(i.e.
has a
separable predual)
iff
there exists
a
faithful
normal representation
of
93ί
on
a
separable Hubert-space. Indeed,
let π be a
faithful
normal repre-
sentation
of
931
on a
separable
Hubert-space
§.
Then
π(93l)
is a von
Neumann
algebra
on
§.
Let
ψeπ(9K)
#
,
then
the
transposed
map
'π
of
π is
defined
by
(
r
π(φ),
A) =
φ(π(A))
for all A ε m.
XTT^)*)
is the
image
of
π(9K)
ϊK
in
93?^
because
π is
normal. Furthermore every
φ ε
931^
is
the
image
of
some
ipεπ^)^.
under
the
transpose
*π
of π.
Indeed:
φ(A)
=
(φ
°
π~
1
)
(π(^4))
=
fπ(φ
° π~
*),
^4),
v4
e
931,
where
φ °
π~
1
is
normal
because
φ is
normal
and π is
faithful
normal. Therefore
we
have that
XπίSWy
=
931*.
In
addition
we
have that
|Γπ(tp)||
=
II
Vli
for all
ψ
ε
71(931)*
namely:
l
= sup
|('π(
V
M)l
= sup
\ψ(π(A))\
=
\\ψ\\
.
We
thus conclude that
931*
is
isometric
as a
Banach space
to
π(93ί)*
which
in
turn
is
isometric
to
β(§)*/π(93l)
1
,
where
#(§)*
is the set of
trace-class operators
on
§
and
π(93l)
1
is the
annihilator
of
π(93ί)
in
£(§)*.
Since
§
is
separable
and
π(9K)
1
closed,
β(§)*/π(93ϊ)
1
is
separable
and
consequently
π(93l)*
(and
hence
931*)
is
separable.
If we now
conversely
consider
the
situation where
93Ϊ*
is
separable, then there exists
a
countable
dense
set
{φ
n
}
in
93Ϊ*.
From this
one
constructs
φ = Σ
———
Γ
φ
n
,
then
φ
is
normal,
as a
norm-limit
of
normal linear positive forms, furthermore
φ is
faithful
on
931
and
gives rise
to a
faithful
normal cyclic representation
π
φ
of
931
on a
separable Hubert space
§
φ
.
The
separability
of
§
φ
follows
from
the
fact
that
the
σ(93ϊ,
931*)
topology
on the
unit ball
of
931
is
metrizable
and
therefore
we can
find
a
countable
σ(93ί,
93Ϊ*)
dense subset
of
93Ϊ,
{x
n
}
say.
Let
Ω
φ
be the
cyclic vector
for
π
φ
(93ϊ)
in
ξ)
φ
.
For
every
ξεξ)
φ
with
the
property
(£,
π
φ
(xj
Ω
φ
)
=
OVx
n
e
9W,
we
have that
(ξ,π
φ
(x)Ω
φ
)
=
OVxe93ί
because
ω
ξ
Ω
is
σ(93l,
93Ϊ*)
continuous
and
π
φ
is
normal.
Because
Ω
φ
is
cyclic
for
π^(93l)
we
conclude that
£
=
0, and
therefore
that
{π
φ
(x
n
)
Ω
φ
}
is a
dense
set of
vectors
in
§
φ
which then
is
separable.
A
PF*-algebra
931
is
σ-finite
iff
there exists
a
faithful
normal state
on
931
[8].
From what
was
said above,
it
follows that every separable
PΊ^*-algebra
is
σ-ίϊnite.
A
particular case
is
when
the
algebra
U
could
be
thought
of as
generated
on
Fock space.
The
funnel
then consists
of von
Neumann
algebras
on
Fock space
and
because this space
is
separable
every
component
93ί
α
in the
funnel
is
also separable
and of
course
σ-finite.
Let U be a
C*-algebra with
a
σ-finite,
properly
infinite
funnel
then
for
every
93l
α
and
931^
with
93l
α
g93ί^
the
embedding
is
normal
([34],
Theo-
rem 7).
132
M.
Takesaki
and M.
Winnink:
A
locally normal linear
form
on U is an
element
from
U*,
with
the
property that
its
restriction
to
every
ΪR
α
is
normal
(i.e.
σ(9K
α
,
(SERJJ
continuous).
Because
the
embeddings
ϊ^gϊR^
for all
ordered pairs
in
the
funnel
are
normal
we can
apply
[12]
(Proposition
6) and
conclude
that
the
locally normal positive
forms
on U
form
a
folium
(i.e.
they
form
a
uniformly
closed convex subset
in
U*,
with
the
property that
for
every
φ in
this
set
φ
A
(-)
=
φ(A*
-
A)
also belongs
to
it).
We
shall nowhere
assume that
we
have
a
full
folium.
Let U be a
C*-algebra
with
a
sequential
separable
funnel.
Then
in
every
$)l
n
in the
funnel
there exists
a
separable C*-algebra
U
n
that
is
σ(9H
w
,
(9KJ
J
dense
in
9K
n
.
Consider
the
separable C*-algebra
U
0
-
(J
<&»
.
n
Let
V
denote
the set of
locally normal linear
forms
on U. Let S
denote
the
set of
states
on U; the set of
locally normal states
is
then
SnF.
By
σ(Sn
V,
U) we
denote
the
topology induced
on
Sn
V by the
σ(U*
U)
topology (i.e.
the
weak*-topology)
on U*. By
cr(Sn
V,
U
0
)
we
denote
the
locally
convex topological structure induced
on
Sn
Fby
the
semi-norms
on
V: φ
->
|φ(^4)|,
φ
e
F, A
e
U
0
.
We
then obviously have that
σ(6n
V,
81)
<σ(8n
V,
H
0
)
.
Let
K be any
σ(6nF,
U)
compact subset
of
SnF.
Whenever
we can
show that
σ(SnF,
U
0
)
is a
Hausdorff topology then
σ(SnF,
U) and
σ(SnF,
U
0
)
coincide
on K
[23].
In
that case
we
have, because
U
0
is
a
separable C*-algebra, that
K is a
metrizable compact
in
σ(Sn
F, U).
This
then
is
independent
of the
particular
choice
of
U
0
.
In
order
to
show
that
σ(Sn
V,
U
0
)
is a
Hausdorff topology
it
suffices
to
show that
er(F,
U
0
)
is
Hausdorff because
σ(SnF,
U
0
)
is the
topology induced
on
SnF
by
σ(F,U
0
).
Let
φeF
and
φφO,
then
φ(^)
=
OV^eU
0
implies
φ(x)
=
OVxe9K
n
,
n is
arbitrary. This
is
because
50ϊ
n
is a
separable
von
Neumann
algebra
and we can
find
a
separable C*-algebra
U
n
which
is
σ(yjl
n
,Wl
n
^)
dense
in
S0ϊ
n
.
Indeed,
let
x
e
9W
n
then there exists
a
sequence
x^
in
lί
n
such that
x
n
κ
-*x
σ(9M
n
,9Jl
nHe
)
weakly. Since
we
assumed
φ(yl)
=
OV^4eU
0
and φ is
normal
(i.e.
σ(SDl
n5
SDl
ΛJ(c
)
continuous)
on
9Jί
π
we
find
that
<jp(x)
=
OVxeSER
n
,
n
arbitrary. Therefore
φ
vanishes
on
|J
9Jl
w
and
because
φ e
U*,
φ has to
vanish
on all of U,
which
is in
n
contradiction
with
the
assumption
φφO.
In
other words
to
every
φ
e
F,
φ Φ 0
there exists
an
element
AeU
0
with
the
property that
φ(A)
Φ 0,
implying that
σ(V,
U
0
)
is a
Hausdorff
topology.
§
3.
K.M.S. States
in
Connection with
Local
Normality
Let U be a
C*-algebra
and t
-+
σ
t
a
representation
of the
additive group
of
the
real numbers into
the
group
of
*-automorphisms
of U.
Quantum Statistical Mechanics
133
A
state
on lί
satisfies
the
K.M.S. boundary condition,
at the
inverse
temperature
/?,
whenever
the
following
two
conditions
hold:
i)
ω(Aa
t
B)
is a
continuous
function
of t for all A, B e lί.
ii)
$ω(Aσ
t
(B))f(t
-
iβ)
at
=
$ω(σ
t
(B)
A)f(t)
at
for
every
/eT>,
where
T) is the
space
of the
Fourier-transforms
of
functions
in
D.
[Equivalently
ω
satisfies
the
K.M.S. boundary condition, whenever
for
each pair
A and B in lί
there exists
a
holomorphic
function
F in
0 <
Imz
< β
with boundary values
F(ί)
=
ω(σ
t
(A)
B) and F(t + iβ)
=
ω(Bσ
t
(A)\~]
For
details
on the
structure
of
representations induced
by a
K.M.S. state
we
refer
the
reader
to
[11,
37,
40,41].
A
well known
fact
by now is
that every K.M.S. state
ω as
defined
above,
gives
rise, by the
GeΓfand-Segal construction,
to a
representation
π
ω
of lί
with
the
property that
π
ω
(lί)"
has a
cyclic
and
separating vector.
π
ω
(U)"
is
therefore
a
σ-fmite
von
Neumann algebra
[8].
Theorem
1. Let lί be a
C*-algebra
with
a
σ-finite properly
infinite
funnel,
and ω a
K.M.S.
state
on lί
then
ω is
locally normal.
Proof.
Let
(π
ω
,
§
ω
,
Ω
ω
)
be the
representation,
the
Hubert-space
and
the
cyclic vector obtained
from
the
GeΓfand-Segal construction.
As
already remarked
Ω
ω
is
separating
for
π
ω
(U)"
and
therefore
π
ω
(U)"
is
σ-finite.
Let
9Jί
α
be a
component
of the
funnel,
then
π
ω
|9Jl
α
is a
*-homo-
morphism
of a
σ-finite
VF*-algebra
into
the
σ-finite
algebra
π
ω
(U)".
Such
a
mapping
is
normal
([34],
Theorem
7).
Hence
ω is
locally
normal,
q.e.d.
Remark. Since [34] Theorem
7
also holds
for
every *-homomorphism
of
a
finite
factor
into
a
σ-finite
von
Neumann-algebra,
we see
that
ω
is
normal
on
every
finite
factor
in lί and
also
on
every
σ-finite
properly
infinite
W*-algebra contained
in lί
whether
or not
contained
in the
funnel.
Of
course Theorem
1
holds also
if the
funnel
is a
finite-factor-funnel.
If we
specialize
to a
sequential separable properly
infinite
funnel
2
then
we
have
the
following.
Corollary
1.
Every K.M.S.
state
on a
C*-algebra
lί
with
a
sequential
separable properly
infinite
funnel,
is
locally
normal,
acts
on a
separable
Hubert-space
and is
normal
on
every
factor
contained
in lί.
Proof.
From Theorem
1 we
know that
π
ω
19Jl
n
is
normal.
$R
n
is
sepa-
rable therefore
[π
ω
991,,
Ώ
ω
]
is a
separable subspace
of
§
ω
and
thus
$
ω
(=
\J
(πjaWJΩβΛ
is
separable.
By
[10,17,
35] we
know that every
V
n )
representation
of a
factor
on a
separable Hubert-space
is
normal. q.e.d.
2
Everywhere
in the
following
the
reader
can
substitute
for
each properly
infinite
H^-algebra
a
finite
factor
without changing
the
results.