It is well known that any derivation on a commutative von Neumann algebra is implemented by a bounded operator. This result can be extended to certain ...
BULL. AUSTRAL. MATH. SOC. VOL.
47B47
32 (1985), 415-418.
DERIVATIONS ON COMMUTATIVE OPERATOR ALGEBRAS MARK SPIVACK
It is well-known that any derivation on a commutative von Neumann algebra is implemented by a bounded operator.
In this note we
present a simple alternative proof, which generalises the result further within Hilbert space, and to reflexive Banach spaces.
It is well known that any derivation on a commutative von Neumann algebra is implemented by a bounded operator.
This result can be extended
to certain non-self-adjoint commutative algebras by similarity.
In this
note we present a simple alternative proof, which generalises the result further within Hilbert space, and to reflexive Banach spaces. Let us call a set of Banach space operators a projection algebra if it is the algebra, or norm-closed algebra, generated by its projections. We will say that a projection algebra is bounded, with bound k , if the real number
k
is a uniform bound for the projections in the algebra.
If
E
is a Banach space, then
on
E
and a derivation on a subalgebra
into
B(E)
such that
B(E)
denotes the set of bounded operators A
is a linear map
6(ab) = a& (b) + 6{a)b for
a,b
in
derivation is said to be implemented if it is of the form for some fixed operator
x
on
E
and all
a
in
6
from A
A . A d (a) = xa - ax
A . All Banach spaces
mentioned are assumed to be complex, and all algebras to be unital. that any "complete" Boolean algebra of projections is bounded.
Note
(See [2]).
Every derivation on a commutative algebra generated by a finite set of Banach space projections is known to be implemented by an operator
Received 11 April 1985. Copyright Clearance Centre, Inc. Serial-fee code: $A2.00 + 0.00. 415
0004-9727/85
416
Mark Spivack
which can be described explicitly.
We obtain a bound for this operator
which is independent of the number of generating projections.
It follows
immediately that continuous derivations on commutative bounded projection algebras on reflexive Banach spaces are implemented. We abuse notation a little by saying that projections orthogonal if pq = qp = 0 . Denote
p and q are
1 - p by p
Then, given any finite commutative set {q .},i = 1,...,n of projections, there is a set {p . },i = 1,...,2
, of orthogonal projections
"2-
(some of which may be zero) generating the same algebra as
{q.} .
Before proving the c r u c i a l result we mention the fact that a Banach space
E
i s reflexive i f and only i f the unit b a l l of
B(E) i s compact
in the weak operator topology (Chapter VI, Dunford and Schwartz [Z]). Note also that a von Neumann algebra on a Hilbert space i s generated by i t s s e l f - a d j o i n t projections [?]• We f i r s t s t a t e as a lemma the known f i n i t e case. LEMMA 1. algebra {p.}, x =
Let
A be a finitely
on any Banach space
generated
E and let
i = l , . . . , n j is an orthogonal n £
&ip-)p-
implements
commutative
projection
6 be any derivation
generating
set,
on A.
If
then the operator
6 on A.
The proof follows easily from the relations p6{p)p = 0 for orthogonal projections THEOREM 2. Hilbert
1 m x = — u) m
~
n I
p6iq)
IIall < 4fc2ll6ll
where
6 (u )u , where for each r r wh
ere each
-6(p)q
k is a bound for
{p.} are self-adjoint,
We claim t h a t for some integer
* • J? •
=
p , q .
space and the projections
Proof.
u
and
X.
1 or
If then
E is a llxll < II6II
m we can write
r , I u II < 2k . r
is
A.
-1 .
In fact we w i l l put
417
Operator algebras Put
m = 2n~1 , and l e t 1
1
{X. } form t h e ( f i n i t e )
1
1-1
1
1
1-1
1
1
1-1
1
.
.
1-1
1
1-1-1 1 1-1-1
1
1
1
1-1-1-1-1
One can easily convince oneself, by checking with m y
and
X.
X.
= 0
n = 2 or 3 , that
for i ? j .
• r 2 I (X. ) = m f o r each ^ . C o n s e q u e n t l y , r=X ' m m n T 6 ( u )u = I I X . X . 6{p .)p . v
r=l
r
z
ij =
m 1
Finally, since each A and u^ = 1 .
X. t,r
r
°r
v
°
n 1 &(p-)p- = m r , as required.
r=l i=l
in
matrix
is ±1 , u r
V
%
is the difference of two projections
DM^I < 2/C , and IIdl < (2fe)2ll6ll .
Hence
i s a s e t of s e l f - a d j o i n t H i l b e r t space p r o j e c t i o n s then each s e l f - a d j o i n t u n i t a r y o p e r a t o r , and so
IIU II = 1 .
Then
If u
Uxll < II6II
{pA is a as
claimed. We can immediately s t a t e t h e c o r o l l a r y .
COROLLARY 3 . Let A be any commutative on a reflexive A . where and
Then
Banach space
E , and let
6 be a continuous
6 is implemented by an operator
k is a bound for A . A a von Neumann algebra
bounded projection x such that
If, in particular, then we can choose
algebra
derivation
on
llxll < Ak II6II ,
E is a Hilbert x so that
space
II xll < II6II .
Proof. The restriction of 6 to any finitely generated projection subalgebra B i s implemented by such an operator, by Theorem 2. Let 5 D D
denote the set of operators which implement
bounded by 4k II6B .
Then
6 on B and whose norms are
5 S R i s compact in the weak operator topology, B
418
Mark Spivack
since
E
empty.
is reflexive, and any finite intersection of such sets is non-
Hence the intersection of all such sets is non-empty, and the
result follows. REMARK 1.
The condition on E of reflexivity applies of course to
Hilbert space, where the self-adjoint projections are exactly those of norm 1.
Any projection on a Hilbert space which commutes with its adjoint
must be self-adjoint.
So a commutative projection algebra on a Hilbert
space is self-adjoint if and only if it has bound 1. holds for general commutative projection REMARK 2.
Note that the result
C -algebras.
We mention finally that this proof is closely related to
an existing one for the case of a von Neumann algebra.
I am indebted to
Professor Erik Christensen for describing that one to me. here for comparison: algebra
Let 6 be a derivation on a commutative von Neumann
A . Note that by Ringrose [3] 6 is continuous.
weakly closed convex set generated by elements of the form u
We sketch it
is unitary in A . For each unitary
T (a) = uau* + uS{u*) . Then
C
u define a map
and the maps
T
Let C be the u
