135
Internat. J. Math. & Math. Sci.
(1990) 135-138
VOL. 13 NO.
A NOTE ON ONE-PARAMETER GROUPS OF AUTOMORPHISMS
A.B. THAHEEM
AND
NOOR MOHAMMAD
Department of Mathematics
Qulad-i-Azam Unlverslt y
Islamabad, Pakistan
(Received July 7, 1988 and in revised form September I, 1988)
Let
ABSTRACT.
{t :t
*-automorphisms of a
t R.
central
The purpose of
and
{Bt:t
be two commuting one-parameter groups of
Bt B-t
a +
+
t
-t
a
this note is to provide
simple and short
for all
proof of the
t
on Mp and a
result:
B_t on M(l-p) for
t
t
M, without using the theory of spectral subspaces.
p
WORDS
R}
Neumann algebra M such that
decomposition
projection
KEY
R}
yon
AND
PHRASES.
Automorphlsm
groups,
central
a
projections,
central
invarlant
projections.
1980 AMS SUBJECT CLASSIFICATION CODES.
1.
Primary 46LI0, Secondary 47C15
INTRODUCTION
In [I], A. Van Daele, L. VanheeswiJck and one of the authors proved that if
*-automorphlsms of avon Neann algebra M satlsfylng the equation:
t -t
+
=t
+
=-t
R, then M can be decomposed nto subalgebras Mp and M(l-p) for
a central proJectlon p in M such that
on M(l-p), tR.
on Mp and
t
e proof of thls result is very technical and fairly long. It depends on veson’s
theo of spectral subspaces ([2], [3]) and as such it lacks a proper emphasis on the
decomposition itself.
However, there are Important sltuatlons where It Is enough to
consider the comtIng automorphlsm groups (see, for Instance [4], [5]). e purpose
for all t
t
8t
of thls note Is to provide a simple proof of thls decomposition result for comtlng
automorphlsm groups without using the theory of spectral subspaces.
We use simple
In dolng so we lose the
a
the
we
For more
hand
of
the
result
but
other
on
simple proof.
get
generality
details concerning the origin of thls operator equation, Its applications and related
dec6osltlon results, we refer to [6], [7], [8], [9], [I0], [4] and [5].
algebralc technlques to obtain the proof.
Of
course,
A.B. THAHEEM AND N. MOHAMMAD
136
2.
MAIN RESULTS
The following is an important decomposition result for comnmtlng automorphisms
([i0], [ii]).
M
+
-I
a
8 on
Mp
such that
such that
a
a
a,B be commuting *-automorphlsms of a yon Neumann algebra
Then there exists a central projection p in M
+ -I
-I
and a
B on M(l-p).
Let
PROPOSITION 2.1.
In this paper, however, we use another version (Proposition 2.2) of the above
result for the sake of clarity.
The essential idea is that when we consider the
aB on M, then by [12], N(aB- I) + R(aB-1) is o- weakly dense in
N(aB-I) and R(aB-I) denote, respectlvely, the null space and the range
R(aB-I) c N(a- B) and the subalgebra
space of the operators under consideration.
L (say) generated by R(aB-I) is a two-sided ideal in M and there exists a largest
central projection
in M such that L
c_ N(a- B) and P0 is a,
*-automorphism
M
where
MP0 MPo
P0
B-
a(po)
invarlant (that is
(see for instance [10],
B(po)
[II]).
i
(note that L c_ N(aB- I))
B-I
M such that a
(I
on
qo
p0)(l
Mqo
B
po ). In other words, a
by considering
Similarly,
on Mp
0
the orthogonal
we get a largest a,B- Invarlant central projection
q0
in
and the orthogonallty relation implies that
For more details, we may refer to [10] and [II].
0
i
ideal L
Thus we may
have the following alternative version of Proposition 2.1.
Proposition 2.2.
-I
such that a + a
that a
on
projection
PO
Mp
Let a, 8 be commuting *-automorphlsms of avon Neumann algebra M
-I
Among the projections p eM (respectively q eM) such
B + 8
in M (respectively
invarlant and (I
-I on
(respectively a
qo
p0)(l
qo
in M) such that
{at:t
*-automorphisms
all t eR.
at(p)
of
a
there
P0
and
exists
q0
a
are a,
largest
B-
yon
{t:t
e R} be two commuting one-parameter groups
Neumann algebra M such that at + a
-t
8t + -t
Then there exists a central projection p in M such that
t(p)
PROOF
p for all teR and
a
t
By Proposition 22, let
-invariant central projection in
a
2-n
8
2-n
central
O.
We now come to our main result.
e R} and
THEOREM 2.3. Let
of
Mq)
8
on
Pn’
M
Mp
8
t
on Mp and a
t
neN, be the largest
such that
n
-t on M(l
p).
for
137
ONE-PARAMETER GROUPS OF AUTOMORPHISMS
Similarly, we let
qn’
n N
be the largest a
o-
in M such that
B
x
-n
2
Then (I
Put p
a
and
MPn+l
lim
n
-m
-m
2
m e N. The
2
that p is
by
m
{k2-
m:
on
MPn+l
maximality
t’
of
the
m
(x) for x
Pn"
set {k
2-
m:
Mp c_C
MPm"
B-t
on
Mq.
As (I
Z, m
k
R.
0.
Since
2- m’ 2-
Similarly q is
at,
Pn
m
N] in R and
2-n
{qn is
decreasing.
is
-invariant for any
the
Bt
coinclde
continuity imply
-invariant.
Also
Again by the density of the set
k Z, mN}, we conclude that a
t
is decreasing because
and
implies that their squares x
n
2
Similarly the sequence
Pn+1 (
3 t -invariant for any t
k2-
Similarly,
{pn}
and q
density
-invariant central projection
Mq n
lira qn" Then clearly (I-p)(1-q)
n
-invariant for n
m, it follows that p is
Pn
(x)
k2-
on
n
0 for any n 6 N. The sequence
2- (n+l)
(n+l)
2on
-pn)(1 -qn
2-
n’ B 2- n
t
Bt
q)(l
p)
R.
on Mp for any t
0 then M(I
p)
.c_
Mq and this
completes the proof of the theorem.
ACKNOWLEDGEMENTS.
International
Atomic
The
authors would like to thank Professor Abdus Salam, the
Energy Agency and UNESCO for hospitality at the International
Centre for Theoretical Physics, Trieste.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
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DAELE, A. VAN., Arveson’s Theory of Spectral Subspaces, Nieuw Arch. Wisk. 27
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Automorphisms of a yon Neumann Algebra,
+
THAHEEM, A.B., On the Operator Equation a + ,
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ow
8.
& Math. Sci. 9 (1986), 767-770.
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A.B. THAHEEM AND N. MOHAMMAD
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9.
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yon