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Chapter II Locally Compact Semi-Algebras
CHAPTER I 1
LOCALLY COMPACT SEMI-ALGEBRAS
The f i r s t main r e s u l t of t h i s c h a p t e r (Theorem 1 . 1 ) g i v e s g e n e r a l s u f f i c i e n t c o n d i t i o n s f o r a non-zero element i n a Banach a l g e b r a t o g e n e r a t e a l o c a l l y compact semi-algebra.
T h i s g e n e r a l i z e s e a r l i e r results
of B o n s a l l and Tomiuk c41 and t h e a u t h o r s C l g l . The c o n d i t i o n s may be l o o s e l y s t a t e d as f o l l o w s : Apart from a f i n i t e set o f poles which do n o t
l i e on t h e p o s i t i v e real a x i s t h e p e r i p h e r a l spectrum of t h e g e n e r a t o r c o n s i s t s o f a set of p o l e s , one of which i s t h e s p e c t r a l r a d i u s . T h i s r e s u l t i s due t o Kaashoek ( s e e C h a p t e r I of C l ] ) . A semi-algebra
i s s a i d t o be s t r i c t i f t h e sum of any two e l e m e n t s i n
t h e s e m i - a l g e b r a i s z e r o o n l y i f b o t h o f them are z e r o . Theorem 1.2 ( a l s o due t o Kaashoek C171) g i v e s s u f f i c i e n t c o n d i t i o n s f o r a non-zero element of a Banach a l g e b r a t o g e n e r a t e a s t r i c t , l o c a l l y compact semi-algebra.
The
c o n d i t i o n s are t h a t t h e p e r i p h e r a l spectrum of t h e g e n e r a t o r s h o u l d c o n s i s t
of p o l e s and t h a t among t h e s e t h e s p e c t r a l r a d i u s s h o u l d b e of maximal o r d e r . The p r o o f s of Theorems 1 . 1 and 1.2 are long b u t u s e o n l y e l e m e n t a r y s p e c t r a l theory. S e c t i o n s 2 and 3 of C h a p t e r I1 are b a s e d on B o n s a l l ' s work c21 on t h e
structure of l o c a l l y compact s e m i - a l g e b r a s ,
a l t h o u g h i n some cases t h e
p r o o f s are d i f f e r e n t . Local compactness i m p l i e s t h e e x i s t e n c e of minimal c l o s e d two-sided i d e a l s (Lemma 2 . 1 ) m d t h i s , i n t u r n , l e a d s t o t h e e x i s t e n c e o f idempotents (Theorem 2 . 5 ) . One o b t a i n s a t h e o r y c l o s e l y r e l a t e d t o c l a s s i c a l Wedderburn t h e o r y ; t h e f i n a l result (Theorem 3.1) b e i n g a c h a r a c t e r i z a t i o n of t h e semi-algebra o f p o s i t i v e m a t r i c e s of f i x e d
26
SWI-ALGEBRAS
order as a strict locally compact semi-algebra which possesses no proper closed two-sided ideals.
1. DEFINITIONS AND EXAMPLES
Let B be a complex Banach algebra with unit e. A non-empty subset A of
semi-algebm if A is closed under addition, multiplication and multiplication by non-negative real scalars, i.e.,
B is said to be
a
a,b
t
A, a
2
0 ->
a +
b, ab, aa
t
A.
A semi-algebra A is called locally compact if A contains non-zero elements and
A
n {X c
11
B: 11x11
is a compact subset of B. These axioms imply th8.t A is a closed subset of B and that A is a locally compact topological space with respect to the relative topology induced in A by the norm topology of B.
A semi-algebra A is called closed if A is a closed subset of B. The intersection of a set of closed semi-algebras in B is a closed semi-algebra. Thus, if t t B, the smallest closed semi-algebra containing t exists. This set is denoted by A(t). Note that A(t) is the closure in B of the set {alt +
... + aktk: ai -> 0 (i = 1 ,...,k), k
+ 1.
c C
A semi-algebra A is called monothetic if A.= A(t) for eome t in A. In that case we s a y that A is generated by t, and we call t a genemtor of A. A semi-algebra A is c m t a t i v e if ab = ba
(a,b
c
A).
Obviously monothetic scmi-algcbrm are commutative. Let A be a semi-algebra in B. The set Bo = A
-A=
(a
- b: a,b
t
A)
is a real eubalgebra of B. If A contains non-zero elements and B
0
is
finite dimensional, then A is trivially locally compact. The next theorem shows that we can construct examples which are not of this type. 1 . 1 THEORFN. Let t be a m-aem element i n B. &pose that u(t) decaposes i n t o two disjoint closed subsets u1 and a2 such that (i) a1 is a f i n i t e (possibly empal set of poles of t and
LOCALLY COMPACT SEMI-ALGEBRAS
IR+ = 0 ; (ii) either u2 = 0 or them e s i s t s 0 < a o
27
n
1
c
u2 such that
u2 = u(t) n ( a : 111 2 a1 and
u
2
,-I ( a :
1x1
= a}
is a f i n i t e set o f poles of t. Then A( t) i s Zocal Zg compact. PROOF.
The sets o 1 and o are spectral sets of t. Let p. be the 2
spectral idempotent associated vith u e = P1 For any a in A(t), api
6
+
(i =
i
Then
1,2).
PIP2 = 0.
P2’
A(tpi) f o r i = 1,Z and a = apl
+
ap2.
Hence it suffices to shov that, for i = 1,2, either A(tpi) is locally compact o r consists of the zero element only. Since
U,
is a finite (possibly empty) set of poles of t, the closed
subalgebra of B generated by tpl is finite dimensional (Proposition P.8). Hence A(tp,) lies in a finite dimensional subalgebra of B . This implies that either A(tpl) is locally compact o r A(tpl) consists of the zero element only. If u2 = 0 , then A(tp2) = {O). Suppose
U2
# 0 . We shall prove that
t2 = tp2 generates a locally compact semi-algebra. First of all note that r(t2) = a, because o2
= o(t2)
c
o2 u (0)
and a c 02. Further
dt2) n (A:
1x1
-
a)
is a finite set of poles of t and thus of t2. Replacing t2 by a-lt2 ve see that vithout loss of generality ve may assume that r(t ) = a = 1. Take bn in A(t2), and assume that Ilbnll to prove that the sequence {b
2
2
for n =
in the closure of the set {alt2 +
,... .
1.2 We have has a convergent subsequence. Since bn is 1
... + %t2:k
ai -> 0, k
6
+ 1,
2
it suffices to verify this statement for the case that
SPII -ALGEBRAS
28
where a k ( n ) 1
t
2
0 and a k ( n )
= 0 f o r k s u f f i c i e n t l y large. Since r ( t 2 )
Cl(t2), t h e spectral mapping theorem ( P r o p o s i t i o n P . 3 ) i m p l i e s t h a t
By u s i n g compactness arguments and t h e d i a g o n a l p r o c e s s , we o b t a i n an i n c r e a s i n g sequence I n . ) i n t
+
1
such t h a t f o r e a c h k t h e sequence ( 4 , ( n i ) }
converges. By p a s s i n g t o t h i s subsequence, we m a y suppose t h a t
Bk = exists. Observe t h a t
l i m %(n) n
L e t q be t h e s p e c t r a l idempotent a s s o c i a t e d w i t h t h e s p e c t r a l s e t
and p u t s = t 2 ( e
- q).
S i n c e r ( s ) < 1 , we know t h a t IIsnII * 0 f o r n *
+o.
T h i s i m p l i e s t h a t t h e sequence
converges i n t h e norm of B. L e t d be i t s sum. Then t h e f o r e g o i n g shows t h a t
Now bn
= b
n
(e
-
d = l i m bn(e n
- q).
Hence i n o r d e r t o prove t h a t
q ) + bnq f o r each n i n 2'.
{bn) h a s u convergent subsequence, it s u f f i c e s t o show t h a t {b q] h a s a n convergent subsequence. But (bnq] i s a bounded sequence i n t h e c l o s e d s u b a l g e b r a of B g e n e r a t e d by t q. From t h e s p e c t r a l p r o p e r t i e s o f t 2 i t 2 f o l l o w s t h a t t h i s a l g e b r a i s f i n i t e d i mensiona l ( c f . , P r o p o s i t i o n P.8).
So Ib q ) h a s a convergent subsequence, and t h e proof is complete. n Let t b e a non-zero element i n B s a t i s f y i n g t h e c o n d i t i o n s of t h e p r e v i o u s theorem and suppose t h a t A(t)
- A(t)
U(t)
is an i n f i n i t e set. Then Bo =
is not f i n i t e d i m c n s i o n d . For suppose t h a t Bo is f i n i t e
dimensional. S i n c e t
n
t
+, t h i n
Bo for each n i n Z
inplies t h e existence
of a non-zero polynomial q such t h a t q ( t ) = 0. But t h e n U ( t ) m u s t b e
LOCALLY COMPACT SEMI-ALGEBRAS
29
finite (by Proposition P.31, contradicting our hypothesis. Using Theorem 1.1, it is easy to construct examples of monothetic locally compact semi-algebras. We mention EXAMPLE.
Let T be
a
otle
particular one.
compact linear operator on a complex Banach space
E, and suppose that T has a positive eigenvalue. Then the monothetic semialgebra generated by T in the Banach algebra L(E) of all bounded linear operators on E is locally compact. A semi-algebra A in B is called s t n ' c t i f
A
n
(-A)
f01,
i.e., if -x and x in A implies x = C. The next theorem shovs how one may construct strict locally compact monothetic semi-algebras. The result is formulated in terns of the set Pem(t)
{ A c u(t):
1x1
= r(tI1.
1.2 THEOREM. Let t be a non-zero element of B such that Pens( t) is a s e t of poles of t. Further suppose that r(t) is a pole of t of rnarimal order i n Pem(t). Then A(t) is s t r i c t and locally compact.
PROOF. The proof consists of two parts. First of all we consider the case r(t) > 0. The local compactness of A(t) follovs immediately from Theorem 1.1 by taking a l = 0 and 02 = U(t). 1.
Let s = Bt for some 8
0. Then A(s) = A(t), r(s) = Br(t), and s and t have
similar spectral properties. Hence in order to prove that A(t) is strict, we may suppose without loss of generality that r(t) = 1. Then 1 is a pole of t of order no, s a y . Let po be the spectral idempotent associated with po. Then (by Proposition P.7) (e t)""l the spectral set { l } . Put xo
-
xo # 0 , txo = x0' n Let b(t) = alt + + a t with ai 2 0 (i = 1 n mapping theorem (Proposition P.3) implies that
...
,...,n).
The spectral
r(b(t)) = b(1). Further we have b( t )xo = b( 1 )xo = r(b( t )xo. Take s in A(t) n (-A(t)).
We want to show that s
-
0 . Since
s
c
A(t),
SEMI-ALGEBRAS
30
t h e r e exists a sequence {bn) i n A ( t ) such t h a t s = l i m bn and bn = cri(n)t i ( n = 1,2, . . . I
F=,
with a i ( n ) 1 . 0 and ui(n) = 0 f o r i s u f f i c i e n t l y l a r g e . From what we proved
i n t h e previous paragraph it follows t h a t , By t h e continuity of t h e s p e c t r a l radius on commutative
f o r n = 1,2,.,.
s e t s (Proposition P . 2 ) , t h e last equality implies t h a t
sx0 = r ( s ) x o . Since
-6
(1)
a l s o belongs t o A ( t ) , we can repeat t h e argument t o show t h a t -sx 0 = r(-s)xo.
(2)
zzl
But r(-s) = r ( s ) . Hence ( 1 ) and ( 2 ) together imply t h a t
lim n
r(8)
cri(n) = l i m r ( b n ) = r ( s ) = 0 . n
= 0 , and so (3)
From our hypothesis it follows t h a t t h e peripheral spectrum of t i s a s p e c t r a l set of t . Let p be t h e corresponding s p e c t r a l idempotent. Then r{t(e
- p)) <
1 , and so
tn(e
- p ) = { t ( e - p))"
-+
o
* -1.
(n
I n p a r t i c u l a r t h i s implies t h a t t h e sequence { t n ( e
- p))
i s bounded i n B,
and so by (3)
That i s , s ( e
-
p) = l i m bn(e
- p) = 0.
Hence in order to prove t h a t s = 0 ,
it is s u f f i c i e n t t o show t h a t s p = 0. Let Peru(t) =
lAil
Then
r 1.
-
= 1 , A 1. i s a pole of t of order ni say,and ni < no ( i = 0,1,2,...
..., r ) . Let p.
{Ai}.
{A~=I,A~,...,A
Then
1
be t h e s p e c t r a l idempotent associated with t h e s p e c t r a l s e t
p = Po + p , +
... + pr.
Hence it s u f f i c e s t o show t h a t
Put oro(") = 0 f o r each n
for
c
0,1,...
,r and j = 0,1,...
.
and each n i n 2+ Observe t h a t f o r f i x e d
LOCALLY COMPACT SEMI-ALGEBRAS
31
c = 0. Consider t h e complex n and 1and f o r j s u f f i c i e n t l y l a r g e @.(XI) J
polynomi a1
a. =
for
..,I and each n i n Z+ . A simple computation
0,1,.
shows t h a t
I n particular
,. ..,r and each n i n . Note t h a t
f o r & = 0,l
,r
f o r e = 0,1,... Since
xo
and each n i n
and j = 0,1,2,...
= 1 i s a p o l e of t of o r d e r n Po, ( t
-
Xo)po,
a r e l i n e a r l y independent. Further
Now s p
0
(4)
+ i3
0'
(5)
+ Z .
t h e elements
... , ( t - x F 0 - l Po n ( t - xo) Op0 = 0 , and t h u s by
(6)
0
(4)
= l i m bnpo. Hence
where
-
..,n
-1).
But t h e elements ( 6 ) a r e l i n e a r l y
Note t h a t ( 5 ) implies t h a t BT)(-)
Also
> 0 f o r j = 0,1,.
J
-13
-1.
0
i n A ( t ) . So r e p e a t i n g t h e argument we see t h a t
f o r some yo,
0 ( j = O,l,...,n
J
0
independent, t h e r e f o r e ( 7 ) and ( 8 ) t o g e t h e r imply t h a t BT)(-) J - 1 , and hence by (5)
j = O,l,...,n
0
e
l i m Bj(n) = for
(t
e= -
n
O,l,...,r
Since h he)"'pL
e
and j = 0,1,...
o
(9)
,n -1. 0
i s a pole of t of order ne with ne'n
= 0. Hence (4) implies t h a t
= 0 for
0
, we
have
SEMI-ALGEBRAS
32
t1
for
. .,r and each n i n Z+ , But then w e can use ( 9 ) t o show t h a t
0,1,.
spa. = l i m bnpr = 0 n
(1 = O , l ,
..., r ) .
This completes t h e proof of p a r t 1 .
u ( t ) = {O), and t h e l o c a l
2 . Next we consider t h e case r ( t ) = 0. Then
compactness of A ( t ) follows immediately from Theorem 1 . 1 by taking Ul
=
U ( t ) and U2
= 0. Observe t h a t t h e assumption r ( t ) = 0 together with
the hypotheses of t h e theorem implies t h a t t i s nilpotent. Let no be t h e order of nilpotence. If no = 1 , then t = 0 , contradicting t h e hypotheses of t h e theorem
Thus no
2 2.
Take s i n A ( t ) n ( - A ( t ) ) . We want t o show t h a t s = 0. Let t h e sequence be as i n p a r t 1 . Then
{bn) i n A ( t
...). no- 1 Using t h e l i n e a r independence of t h e elements t , t 2 , .. . ,t we see t h a t bn =
lyzl-1
ai(n)t
s
where
l i m ai(n)
Ui(m)
=
-
> 0 (i = 1,.
i-
( n = 1,2,
Ql-1
0 for i = 1
repeat t h e argument t o show t h a t
f o r some 6
i
i
ai(4t
,. ..,n
-1.
0
,
(10)
Also -s i n A ( t ) . So w e can
..,n ) . Since t h e elements t , t2,...,tn -1 -1
0
are
l i n e a r l y independent, ( 1 0 ) and ( l l ) together imply t h a t ai(=) = 0 f o r
i
l,.,,,n -1. Thus s = 0. This completes t h e proof. 0
One of our aims is t o get converses of t h e preceding two theorems. For Theorem 1 . 1 t h i s w i l l be t h e main t o p i c of Chapter 111. The results of Chapter 111 a r e based on t h e existence of idempotents i n l o c a l l y compact semi-algebras, a subject we w i l l be dealing with i n t h e next section.
2. IDEMPOTENTS
I n t h i s section B denotes a complex Banach algebra with u n i t element e. L e t A be a semi-algebra i n B. A non-empty subset J of A i s c a l l e d a r i g h t
ideal of A i f ( i ) J i s a semi-algebra,
( i i ) x c A, a E J implies
J. Similarly one defines a l e f t i d e a t . The set J i s s a i d t o be a two-sided BX f
LOCALLY COMPACT SEMI-ALGEBRAS
33
ideat i f J i s both a l e f t i d e a l and a r i g h t i d e a l . An i d e a l ( l e f t , r i g h t or two-sided) J of A i s c a l l e d a closed i d e a l i f J i s a closed subset of A with respect t o t h e r e l a t i v e topology induced i n A by t h e norm topology of B. Note t h a t closed i d e a l s i n a l o c a l l y compact semi-algebra are closed
subsets of B. A closed r i g h t i d e a l J of A i s c a l l e d a minimal closed right ideal i f J
# ( 0 ) and t h e only closed r i g h t i d e a l s of
A contained i n J are ( 0 ) and J.
Similar d e f i n i t i o n s apply f o r minimal closed l e f t o r two-sided i d e a l s .
I n a locally compact semi-algebra each non-aem closed right ideal contains a minimal closed r i g h t ideal. A similar statement holds for $ e f t and two-sided ideals. 2.1 LEMMA.
PROOF.
L e t J be a non-zero closed r i g h t i d e a l of t h e l o c a l l y compact
semi-algebra A. Further l e t W be t h e family of a l l non-zero closed i d e a l s of A contained i n J. The s e t W i s non-empty, and W i s p a r t i a l l y ordered by the inclusion r e l a t i o n
c.
The ordering on W i s inductive. To see t h i s , l e t
V be a non-empty l i n e a r l y ordered subset of W. Put
v,
= {I n ( X E B:
llxll = 1 ) :
I
E
v).
Then V 1 i s a family of compact sets and V 1 has t h e f i n i t e i n t e r s e c t i o n property. Hence n V 1 # 0 and thus Jo = n V contains non-zero elements. Note t h a t Jo i s a closed r i g h t i d e a l of A contained i n J. Hence Jo i s a lowerbound f o r V i n W. So W i s inductively ordered, and w e can apply Zorn’s lemma t o show t h e existence of a minimal closed r i g h t i d e a l contained i n J. The “minimum condition” on t h e closed i d e a l s proved above i s t h e main t o o l i n t h e s t r u c t u r e theory of l o c a l l y compact semi-algebras. I n t h i s s e c t i o n we develop t h i s theory a s f a r as i s necessary t o g e t s u i t a b l e conditions ensuring t h e existence of idempotents. L e t D be a non-empty subset of t h e semi-algebra A. The right annihita-
t o r D of D i s t h e set of a l l x i n A such t h a t r
D~ = {X E A:
BX
=
o
(a
BX E
= 0 f o r each a i n D. Thus D)).
A l e f t annihilator i s defined s i m i l a r l y .
Observe t h a t Dr i s a closed r i g h t i d e a l of A. If D i t s e l f i s a r i g h t i d e a l , then it i s e a s i l y seen t h a t D
r
i s a closed two-sided i d e a l .
34
SEMI-ALGEBRAS
Let E be a closed subset of the locally compact semialgebra A and SUppose that aE c E ( a 2 0 ) . Let a be an element i n A such that 2.2 LEMMA.
{a)r n E = (0).
Then aE
(1)
Closed.
$8
PROOF. Let y = l i m axn, where {xn ) i s a sequence i n E. F i r s t l y , suppose t h a t t h e sequence Ex 1 i s bounded. The l o c a l compactness of A n implies t h e existence of a convergent subsequence { x ) with l i m i t xo, say. "i Then y = axo, and xo E E , since E i s closed. Hence y E aE, which i s t h e desired r e s u l t .
II~~I~I
Next suppose (x-1 t o be unbounded. Then t h e r e e x i s t s a subsequence
2
{xn I such t h a t i
. . . put
i f o r i = i,2,.
x.I = IIxn l l - l x 1 i "i
( i = 1,2,
...1.
Note t h a t l i m ax( = 0. The hypothesis on E implies t h a t xf E E f o r each i . Since
IIxiII
-
= 1 , we can repeat t h e argument of t h e first p a r t of t h e proof
t o show t h a t { x i ) has a limit point xd i n E and ax
I
0
0. By ( l ) , x:
= 0.
However, t h i s contradicts t h e f a c t t h a t IIx
i s not unbounded.
An idempotent p i n A is c a l l e d a r i g h t minimal idempatent of A i f PA
i s a minimal closed r i g h t i d e a l . Note t h a t minimal idempotents are non-zero. 2.3 PROPOSITION.
Let A be a locally compact semi-algebra, and l e t M be a minimat c b s e d e h t idea1 i n A. Suppose that p is a no??-zero idempotent i n M. Then pm = m
(m
E
MI.
In p a r t h Z a r pA = M, and thus p i s a right minimal idempotent. PROOF.
Put J = {m
E
M: m
- pm c MI.
Then J i s a closed r i g h t i d e a l
contained i n M. Since 0 # p E J and M i s m i n i m a l , we have J = M. Each element m
- pm
E
{pjr. Hence t h e foregoing implies t h a t
m
- pm
E
{pJr n M
(m
E
M).
The set {p), n M i s a closed r i g h t i d e a l contained i n M. Since p
4 {PI
r'
LOCALLY COMPACT SEMT-ALGEBRAS
{ p l r n M # M. Hence {p), n M = ( 0 ) , and thus m = pm f o r each m
35 E
M. The
remaining a s s e r t i o n s are t r i v i a l consequences of t h i s r e s u l t . 2.4 PROPOSITION.
Let A be a l o c a l l y compact semi-algebra, and l e t M be a minimal closed r i g h t ideal i n A. Take a i n A and suppose that aM # (0). Then eM is a minimal closed r i g h t i d e a l . The set {a)
PROOF.
r n M i s a closed r i g h t i d e a l of A contained i n M. From our hypotheses it follows t h a t (a), n M # M. Since lul i s minimal, t h i s implies t h a t {a),
n M =
(0).
So we can apply Lemma 2.2 t o show t h a t aM i s closed.
Clearly aM i s a r i g h t i d e a l . It remains t o show t h a t aM i s minimal. L e t I be a closed r i g h t i d e a l contained i n aM, and suppose t h a t I
L e t J = {m
t
# aM.
M: am E I ) . The s e t J i s a closed r i g h t i d e a l contained i n M
and J # M. Hence J = ( 0 ) , and thus I = ( 0 ) . This shows t h a t aM i s a minimal closed r i g h t i d e a l .
Let A be a l o c a l l y compact semi-a1gebm, and l e t M be a m i n k 1 closed r i g h t ideal i n A. Suppose that M2 # (0). Then M contains a non-xem, idempotent. 2.5 THEOR&.
PROOF.
The proof c o n s i s t s of f o u r p a r t s .
1 . From our hypotheses follows t h e existence of an element t i n M such
t h a t t M # ( 0 ) . L e t J = {t), n M. Then 3 is a closed r i g h t i d e a l contained i n M and J # M. Since M i s minimal t h i s implies t h a t J = (0). Hence tx # 0
(0
# x
E
M).
2. Assume t h a t A i s commutative. From ( 2 ) it follows t h a t xM
(2)
# (0)f o r
each non-zero x i n M, and hence we can apply Proposition 2.4 t o show t h a t xM i s a closed r i g h t i d e a l . Further XM c M ( x E M ) . So
xM=M
(OfxcM).
But t h i s implies t h a t M \ 10) i s a group with respect t o multiplication. The unit element of t h i s group i s a non-zero idempotent contained i n M. This completes t h e proof f o r t h e commutative case.
3. Next we r e t u r n t o t h e general case. Let t be as i n p a r t 1. Consider A ( t ) , t h e monothetic semi-algebra generated by t . Since A ( t ) c M , (2)
36
SEMI-ALGEBRAS
implies t h a t t s # 0 f o r each non-zero s i n A ( t ) . But A ( t ) i s commutative. Thus st # 0 f o r 0
# s
E
A ( t ) . Hence
s M # (0)
(0
# s
A(t)).
E
Repeating t h e arguments o f p a r t 1 f o r ' s i n s t e a d o f t we see t h a t sx # 0
(0
#
x
E
M).
I n p a r t i c u l a r s 1 s 2 # 0 f o r each p a i r s 1 and s2 of non-zero elements i n A ( t 1.
4. Observe t h a t A ( t ) i s a commutative l o c a l l y compact semi-algebra. According t o Lemma 2.1, A ( t ) contains a minimal closed i d e a l N , say. Since A ( t ) has no d i v i s o r s of zero ( s e e t h e conclusion of p a r t 31, w e have N2 # ( 0 ) . But t h e n , by t h e result of p a r t 2 , N c o n t a i n s a non-zero idempotent p.
Since N c A ( t ) c M,
p i s a non-zero idempotent i n M. An element p i n a semi-algebra A i s c a l l e d a unit ei!ement of A i f p # 0 and pa = ap = a
(a
E
A).
A semi-algebra A i s c a l l e d a division semi-algebra i f A has a u n i t element
p and each non-zero element x of A has an inverse y i n A , i . e . xy = yx = p.. Equivalently, A i s a d i v i s i o n semi-algebra i f A* = A \ (0)
i s a group w i t h r e s p e c t t o m u l t i p l i c a t i o n . 2.6 THEOREM.
Let p be a r i g h t minimal idempotent i n the localtg compcct semi-algebra A. Then pAp is a locaZty compact division semi-atgebm. PROOF.
Let D = PAP.
Then D is c l e a r l y a semi-algebra w i t h u n i t
element p. Further D i s a closed subset o f A , s i n c e
D = {x
E
A: x = px = xp}.
The l o c a l compactness of A implies t h a t D is l o c a l l y compact. It remains t o prove t h a t D* = D \ EO) i s a group with r e s p e c t t o
m u l t i p l i c a t i o n . Let pap be i n D*. We s h a l l prove t h a t t h e equation
37
LOCALLY COMPACT SFMI-ALGEBRAS
To show t h i s , it s u f f i c e s t o prove t h a t t h e equation
has a solution i n D*.
i s solvable i n PA.
From our hypothesis it follows t h a t M = pA i s a minimal closed r i g h t i d e a l of A. Since p2 = p I 0
# pap
(pa)M.
E
So w e can apply Proposition 2 . k t o show t h a t (pa)M i s a closed r i g h t i d e a l .
But (pa)M
c
(pA)M = M2 c M.
Since M i s minimal, t h i s implies (pa)M = M. Hence equation (4) has a solution i n M = PA. The f a c t t h a t equation ( 3 ) i s solvable in D* proves t h a t each element i n D* has a r i g h t inverse i n D*. Using a standard argument from elementary group theory, one can show t h a t t h i s implies t h a t D* i s closed under multiplication. Hence it follows t h a t D
2.7 THEOREM. element p. Then
is a group with u n i t element p.
Let A be a strict closed d i v i s i o n semi-algebra w i t h unit
+
A=IRp. 0
PROOF.
Let u
E
A. F i r s t of a l l we show t h a t hp
-u
E
A for
> llull.
Consider t h e s e r i e s
with uo = p. For 1 >
1 1,1 1
t h i s s e r i e s converges i n B. Let bh denote i t s
sum. Since t h e p a r t i a l sums of t h e s e r i e s belong t o A and A i s closed, b
x
E
A. Further we have bh(hp
- u) =
(hp
x p - u = bh' Here x-'
-
u ) b h = p. Hence E
A.
denotes t h e inverse of x i n A.
L e t p = i n f { A : hp closed, pp
-u
L
-u
E
A). Since A i s s t r i c t , p
A. W e s h a l l prove t h a t
y = pp
-u
= 0.
2 0.
Since A i s
38
SEMI-ALGEBRAS
Suppose not. Then, by t h e argument of t h e f i r s t paragraph, vp
- y-1
E
A for
v s u f f i c i e n t l y l a r g e and p o s i t i v e . But then
f o r E > 0 s u f f i c i e n t l y s m a l l . However t h i s c o n t r a d i c t s t h e d e f i n i t i o n of p . Hence y = 0 and u = pp. This shows t h a t A
+
c
+
mop.
Since p
E A
t h e reverse
i n c l u s i o n holds t r i v i a l l y . Thus A = IR p. 0
3. SIMPLE LOCALLY COMPACT SEMI-ALGEBRAS A semi-algebra A i s s a i d t o be simple i f ( 0 ) and A a r e t h e only two-
sided i d e a l s of A. Clearly a d i v i s i o n semi-algebra i s simple. The nxn matrices with non-negative e n t r i e s provide a l e s s t r i v i a l example. Let M ( C ) be t h e a l g e b r a of a l l nxn matrices with complex e n t r i e s n endowed with some a l g e b r a norm. With such a norm'Mn(C) i s a Banach algebra.
+
Denote by M n ( I R o )
t h e semi-algebra of a l l matrices belonging t o M n ( C ) t h a t
have non-negative e n t r i e s . This semi-algebra i s l o c a l l y compact, s t r i c t ,
+
simple and it has a u n i t element. The next theorem shows t h a t Mn(lRo)
is
c h a r a c t e r i z e d by t h e s e f o u r p r o p e r t i e s . 3.1 THEOREM. Let A be a 1ocaZZy compact, s t r i c t , simple semi-algebra w i t h a unit element. Then there e x i s t s a p o s i t i v e integer n such thut A is + isomorphic to the semi-algebra Mn( lRo )
.
The proof goes i n a number of s t e p s . I n t h e sequel A i s a l o c a l l y
compact, s t r i c t , simple semi-algebra with u n i t element 1. Since A # ( 0 ) ,
w e know t h a t 1 # 0. Given non-empty s u b s e t s B and D of A, w e denote by BD t h e set of a l l f i n i t e sums bld, + with bi
3.2
E
B and d.
1
E
. . a
+ bndn
D.
Let M be a nun-zero r i g h t ideal i n A. Then M2 #
PROOF.
(0).
Suppose M2 = ( 0 ) . Then (AM)(AM)
=
AMz = ( 0 )
Note t h a t AM i s a two-sided i d e a l , and AM # ( 0 ) s i n c e 1
(1) E
A. But A i s
LOCALLY COMPACT SEMI-ALGEBRAS simple, t h e r e f o r e AM = A. Then ( 1 ) implies t h a t A' fact that 0 # 1
A*. Hence M2
6
#
39
= ( 0 ), contradicting t h e
(0).
Since A i s l o c a l l y compact, Lemma 2.1 implies t h e e x i s t e n c e of a minimal closed r i g h t i d e a l i n A. By what we have j u s t proved, each minimal closed r i g h t i d e a l i n A s a t i s f i e s t h e conditions of Theorem 2.5. Hence there exist
r i g h t minimal idempotents i n A ( s e e Proposition 2 . 3 ) . Let
denote t h e s e t of a l l r i g h t minimal idempotents i n A . Then 3.3.
TI # 0.
TIA = A.
PROOF.
Observe t h a t n A i s a non-zero r i g h t i d e a l of A. Since A is
simple, it s u f f i c e s t o show t h a t RA i s a l e f t i d e a l , t h a t i s (2)
A ( n A ) c TIA
and a i n A. If ap = 0 , t h e n apx = 0
Take p i n
#
Suppose ap
E
JfA f o r each x i n A.
0. Then ( a p ) ( p A ) # ( O ) , and we can apply Proposition 2.4 t o
show t h a t (ap)(pA) i s a minimal closed r i g h t i d e a l i n A. Each minimal closed r i g h t i d e a l i n A s a t i s f i e s t h e conditions of Theorem 2.5. Hence there exists q
E
TI
such t h a t (see Proposition 2.3) (ap)(pA) = qA.
But then ap = qb f o r some b i n A , and t h u s apx = q ( b x ) E n A f o r each x i n A. Hence ( 2 ) holds.
For p and q i n
3.4. PROOF.
n,
the s e t pAq # ( 0 ) .
Suppose pAq = ( 0 ) . Then q
E
(PA),.
The s e t (PA),
i s a two-
s i d e d i d e a l containing a non-zero element, namely q . Therefore (@), s i n c e A i s simple. However, p Thus pAq
p
4 (PA),.
s
A
Contradiction.
(0).
Given p and q i n II, there e x i s t s a rwn-zero u i n A such that
3.5.
pAq = IR+.
If p = q,
then we may take u t o be p.
PROOF.
Take v = qap # 0 i n qAp. Then v( PA)
( 0 ) and t h u s , by
Proposition 2 . 4 , t h e set v(pA) i s a minimal closed r i g h t i d e a l of A. From v(pA) c qA we s e e t h a t v(pA) = qA, and t h e r e f o r e
SEMI-ALGEBRAS
40
v(pAq) = q k . Hence t h e r e e x i s t s u i n pAq such t h a t vu = q. Observe t h a t (pAq)v = pAqap c PAP.
+
According t o Theorems 2.6 and 2.7, pAp = Bop. So
+
+
0
0
pAq = (pAq)vu c IR pu = IR u. On t h e o t h e r hand u
E
+
+
pAq, and t h e r e f o r e IR u c pAq. Hence pAq = IRou. 0
I f p = q , t h e n by Theorems 2.6 and 2.7 we may t a k e u t o be p. 3.6.
There e x h t s u 1 ,u2,. 1 = u1
PROOF.
i n A ( i = 1, 2 , . .
... + un,
+ u2
Because 1
E A
.. ,un in Jl such that
and
. ,n) such t h a t
nA
u.u = i j
o
( i # j).
= A ( s e e 3 . 3 ) , t h e r e e x i s t ei i n
1 = e a + e a +...+ 1 1 2 2
and a
e a . n n
i
(3)
We may suppose t h a t t h e r e p r e s e n t a t i o n ( 3 ) of 1 is chosen so t h a t n i s as
small a s p o s s i b l e . Postmultiplying ( 3 ) by e l g i v e s el = e a e 1 1 1 By 3.5 we have e l a e l = he (I
Formula
(4) implies
1
... + enan e 1 '
+ e2a2el +
with h
2
0. Hence
- h ) e l = e 2a2e1 + ... + en ane 1 '
that h
2
1, for, if h
(4)
1 , t h e n we can s u b s t i t u t e f o r
e l i n ( 3 ) and we o b t a i n a r e p r e s e n t a t i o n of t h e u n i t element o f t h e form 1 = e2b2
+
,..
+
say, which c o n t r a d i c t s t h e d e f i n i t i o n of n. Therefore h L 1 . On t h e o t h e r hand, because of t h e s t r i c t n e s s of A , formula
(4) shows
that h > 1 is
impossible. Thus h = 1 and hence e ae
= e l , e2a2el +
... + enan e 1 = 0.
Again using t h e s t r i c t n e s s of A , w e s e e t h a t t h e second p a r t of t h e l a s t formula implies t h a t e . a . e
1 1 1
= 0 f o r i = 2,...,n.
By r e p l a c i n g e l by e j i n
t h e preceding arguments, we o b t a i n e.a.e
= e J J ~j y Let ui = e. a. ( i = 1 , ,n) 1 1
e.a.e
i i j
=
o
( i # j).
... . Then we have proved t h a t
LOCALLY COMPACT SEMI-ALGEBRAS
1 = u
# u. 1
Since 0
... + un ,
+
1
= u? = u.( e . a . ) 1
1
1
1
u u = i j
u? = u. 1
, we
1'
41
o
(i # j ) .
# ( a ) , and s o we can a p p l y
have u.( e i A )
P r o p o s i t i o n 2.4 t o show t h a t u . ( e . A ) i s a minimal c l o s e d r i g h t i d e a l . C l e a r l y , u. 1
3.7.
1
E
1
u . ( e . A ) . But t h e n , by P r o p o s i t i o n 2 . 3 , u. 1
1
E
Jl.
For each i and j we have
(5)
u1 . ~ Ju .= R 0 + e1J' ..
where e . . is some non-zero element of u.Au 1J
j'
1
so that
e i i = u. ( i = 1 ,.
.. , n ) ,
S i n c e u.
E
1J
= e e. e ij jk i k ( i , j ,k = 1
,. . , , n ) ,
( j # k).
e. .e = 0 ij kh PROOF.
The eZements e . . can be chosen
TI, we know from 3.5 t h a t t h e r e e x i s t e l e m e n t s e . .
s a t i s f y i n g ( 5 ) . Moreover, a g a i n a c c o r d i n g t o 3.5, we can choose e .
u. f o r i = 1 ,
... ,n.
Observe t h a t t h e c o n d i t i o n e. .e IJ
kh
1J t o be
( j # k)
= 0
i s automatically s a t i s f i e d since uj\ the e
ii
= 0 f o r j # k . I t r e m a i n s t o choose
so t h a t e . e = ei k ' ij i j jk F o r t h e e l e m e n t s el.i w i t h j # 1 , w e t a k e a r b i t r a r y non-zero e l e m e n t s
o f u,Auj. Then ( 5 ) h o l d s f o r t h e s e e l e m e n t s . Next, l e t j
# 1 and k b e
a r b i t r a r y . The l e f t a n n i h i l a t o r (Au ) i s a two-sided i d e a l n o t c o n t a i n i n g k e S i n c e A i s s i m p l e , t h i s i m p l i e s t h a t (A\)L = ( 0 ) . Therefore
yc.
(0)
# e l jAuk = e , jujA\
c ulA\
and hence eljujA\
= ulA\.
T h i s shows t h a t t h e e q u a t i o n eljX =
(6)
lk
h a s one ( a n d no more t h a n o n e ) s o l u t i o n i n u.A \, e j k s a y . We have now
Observe t h a t f o r j = k 2 2 t h e s o l u t i o n jk' 0 and b e l o n g s t o ujA\. Thus ( 5 ) h o l d s .
completed t h e c h o i c e o f t h e e of
( 6 ) is u.. Further e . #
J Jk According t o ( 6 ) . w e have e l j e j k = e l k f o r j
we obtain eljejk = elk
( j ,k = 1
#
1. Since el l e l k = elk'
,. . . , n ) .
SEMI-ALGEBRAS
42
-
Take a r b i t r a r y i , j and k. From eijejk E u.A\ 1
+
Doeik,
= hik. But then jk e l k = e l i ( e i j e j k ) = Xelieik
w e conclude t h a t e. .e IJ
p
Xelk.
This implies t h a t X = 1, and hence eijejk = eik f o r all i , j and k . We a r e now able t o prove Theorem 3.1. Take x i n A. Then x = 1.X.l The map x
+
(hij)y,j=l
=
is t h e desired isomorphism.
NOTES ON CHAPTER I1 1 . I n [ 4 1 Bonsall and Tomiuk have proved t h a t t h e monothetic semi-
algebra generated by a compact l i n e a r operator T which has t h e additional property t h a t 0 < r(T)
B
u(T),
i s l o c a l l y compact. This r e s u l t i s t h e f i r s t version of Theorem 1 . 1 . A second version of the theorem appears i n C191, where it waa shown t h a t t h e condition t h a t T i s compact can be replaced by t h e requirement t h a t the peripheral spectrum of T c o n s i s t s of poles of T. 2. It i s easy t o construct an example of a compact operator T with
r ( T ) = 0 which generates a semi-algebra which is not l o c a l l y compact. Let
el be
t h e Banach space of absolutely summable sequences x = {xn) of
complex numbers, and l e t T on 1, be defined by (Tx), =
n+ n+ 1 X
1
( n = 1,2,...)
Then T i s compact and quasi-nilpotent ( i . e . , r ( T ) l o c a l l y compact ( s e e C191 f o r d e t a i l s ) .
-
. O ) , but A ( T ) i s not
3. Each nilpotent operator generates a semi-algebra which is l o c a l l y compact. Whether a quasi-nilpotent, non-nilpotent l i n e a r operator can generate a l o c a l l y compact semi-algebra i s an open problem.
43
LOCALLY COMPACT SEMI-ALGEBRAS
4. The f i r s t example o f an i n f i n i t e dimensional l o c a l l y compact semia l g e b r a i s given by Bonsall i n C21. It goes as follows. L e t A b e t h e s e t of
real numbers c o n s i s t i n g of t h e closed u n i t i n t e r v a l [0,11 t o g e t h e r w i t h t h e
r e a l number 2 , and l e t A be given t h e r e l a t i v e topology induced from t h e u s u a l topology on t h e r e a l l i n e . Denote by A' t h e c l a s s of real-valued f u n c t i o n s defined on t h e closed i n t e r v a l [0,21 t h a t are continuous, nonn e g a t i v e , i n c r e a s i n g and convex t h e r e . F i n a l l y , l e t A denote t h e c l a s s of f u n c t i o n s on A t h a t a r e t h e r e s t r i c t i o n s t o A of f u n c t i o n s i n A ' . Then A i s
a l o c a l l y compact semi-algebra not contained i n a f i n i t e dimensional a l g e b r a ( s e e [21 f o r d e t a i l s ) .
5. Theorem 1.2 i s r e l a t e d t o work of L. Elsner. I n c111 E l s n e r proved t h e following theorem. L e t T be a compact l i n e a r o p e r a t o r on a complex Banach space E. Suppose t h a t r ( T ) > 0 and t h a t r ( T ) i s a p o l e of T of maximal o r d e r i n t h e p e r i p h e r a l spectrum of T. Then t h e r e e x i s t s a closed and t o t a l cone ( s e e s e c t i o n IV.2 f o r t h e d e f i n i t i o n s of t h e s e n o t i o n s )
E such t h a t TK c
K.
The f a c t t h a t
K is a
K
in
closed and t o t a l T-invariant cone
implies t h a t A ( T ) i s s t r i c t ( s e e Chapter IV f o r more r e s u l t s of t h i s t y p e ) . Hence f o r o p e r a t o r s T of t h i s kind t h e f i r s t p a r t of Theorem 1.2 follows from E l s n e r ' s theorem ( c f . Theorem 8 i n C171). Whether o r not E l s n e r ' s theorem holds f o r o p e r a t o r s T s a t i s f y i n g t h e c o n d i t i o n s of Theorem 1.2 i s a n unsolved problem.
LOCALLY COMPACT SEMI-ALGEBRAS
The f i r s t main r e s u l t of t h i s c h a p t e r (Theorem 1 . 1 ) g i v e s g e n e r a l s u f f i c i e n t c o n d i t i o n s f o r a non-zero element i n a Banach a l g e b r a t o g e n e r a t e a l o c a l l y compact semi-algebra.
T h i s g e n e r a l i z e s e a r l i e r results
of B o n s a l l and Tomiuk c41 and t h e a u t h o r s C l g l . The c o n d i t i o n s may be l o o s e l y s t a t e d as f o l l o w s : Apart from a f i n i t e set o f poles which do n o t
l i e on t h e p o s i t i v e real a x i s t h e p e r i p h e r a l spectrum of t h e g e n e r a t o r c o n s i s t s o f a set of p o l e s , one of which i s t h e s p e c t r a l r a d i u s . T h i s r e s u l t i s due t o Kaashoek ( s e e C h a p t e r I of C l ] ) . A semi-algebra
i s s a i d t o be s t r i c t i f t h e sum of any two e l e m e n t s i n
t h e s e m i - a l g e b r a i s z e r o o n l y i f b o t h o f them are z e r o . Theorem 1.2 ( a l s o due t o Kaashoek C171) g i v e s s u f f i c i e n t c o n d i t i o n s f o r a non-zero element of a Banach a l g e b r a t o g e n e r a t e a s t r i c t , l o c a l l y compact semi-algebra.
The
c o n d i t i o n s are t h a t t h e p e r i p h e r a l spectrum of t h e g e n e r a t o r s h o u l d c o n s i s t
of p o l e s and t h a t among t h e s e t h e s p e c t r a l r a d i u s s h o u l d b e of maximal o r d e r . The p r o o f s of Theorems 1 . 1 and 1.2 are long b u t u s e o n l y e l e m e n t a r y s p e c t r a l theory. S e c t i o n s 2 and 3 of C h a p t e r I1 are b a s e d on B o n s a l l ' s work c21 on t h e
structure of l o c a l l y compact s e m i - a l g e b r a s ,
a l t h o u g h i n some cases t h e
p r o o f s are d i f f e r e n t . Local compactness i m p l i e s t h e e x i s t e n c e of minimal c l o s e d two-sided i d e a l s (Lemma 2 . 1 ) m d t h i s , i n t u r n , l e a d s t o t h e e x i s t e n c e o f idempotents (Theorem 2 . 5 ) . One o b t a i n s a t h e o r y c l o s e l y r e l a t e d t o c l a s s i c a l Wedderburn t h e o r y ; t h e f i n a l result (Theorem 3.1) b e i n g a c h a r a c t e r i z a t i o n of t h e semi-algebra o f p o s i t i v e m a t r i c e s of f i x e d
26
SWI-ALGEBRAS
order as a strict locally compact semi-algebra which possesses no proper closed two-sided ideals.
1. DEFINITIONS AND EXAMPLES
Let B be a complex Banach algebra with unit e. A non-empty subset A of
semi-algebm if A is closed under addition, multiplication and multiplication by non-negative real scalars, i.e.,
B is said to be
a
a,b
t
A, a
2
0 ->
a +
b, ab, aa
t
A.
A semi-algebra A is called locally compact if A contains non-zero elements and
A
n {X c
11
B: 11x11
is a compact subset of B. These axioms imply th8.t A is a closed subset of B and that A is a locally compact topological space with respect to the relative topology induced in A by the norm topology of B.
A semi-algebra A is called closed if A is a closed subset of B. The intersection of a set of closed semi-algebras in B is a closed semi-algebra. Thus, if t t B, the smallest closed semi-algebra containing t exists. This set is denoted by A(t). Note that A(t) is the closure in B of the set {alt +
... + aktk: ai -> 0 (i = 1 ,...,k), k
+ 1.
c C
A semi-algebra A is called monothetic if A.= A(t) for eome t in A. In that case we s a y that A is generated by t, and we call t a genemtor of A. A semi-algebra A is c m t a t i v e if ab = ba
(a,b
c
A).
Obviously monothetic scmi-algcbrm are commutative. Let A be a semi-algebra in B. The set Bo = A
-A=
(a
- b: a,b
t
A)
is a real eubalgebra of B. If A contains non-zero elements and B
0
is
finite dimensional, then A is trivially locally compact. The next theorem shows that we can construct examples which are not of this type. 1 . 1 THEORFN. Let t be a m-aem element i n B. &pose that u(t) decaposes i n t o two disjoint closed subsets u1 and a2 such that (i) a1 is a f i n i t e (possibly empal set of poles of t and
LOCALLY COMPACT SEMI-ALGEBRAS
IR+ = 0 ; (ii) either u2 = 0 or them e s i s t s 0 < a o
27
n
1
c
u2 such that
u2 = u(t) n ( a : 111 2 a1 and
u
2
,-I ( a :
1x1
= a}
is a f i n i t e set o f poles of t. Then A( t) i s Zocal Zg compact. PROOF.
The sets o 1 and o are spectral sets of t. Let p. be the 2
spectral idempotent associated vith u e = P1 For any a in A(t), api
6
+
(i =
i
Then
1,2).
PIP2 = 0.
P2’
A(tpi) f o r i = 1,Z and a = apl
+
ap2.
Hence it suffices to shov that, for i = 1,2, either A(tpi) is locally compact o r consists of the zero element only. Since
U,
is a finite (possibly empty) set of poles of t, the closed
subalgebra of B generated by tpl is finite dimensional (Proposition P.8). Hence A(tp,) lies in a finite dimensional subalgebra of B . This implies that either A(tpl) is locally compact o r A(tpl) consists of the zero element only. If u2 = 0 , then A(tp2) = {O). Suppose
U2
# 0 . We shall prove that
t2 = tp2 generates a locally compact semi-algebra. First of all note that r(t2) = a, because o2
= o(t2)
c
o2 u (0)
and a c 02. Further
dt2) n (A:
1x1
-
a)
is a finite set of poles of t and thus of t2. Replacing t2 by a-lt2 ve see that vithout loss of generality ve may assume that r(t ) = a = 1. Take bn in A(t2), and assume that Ilbnll to prove that the sequence {b
2
2
for n =
in the closure of the set {alt2 +
,... .
1.2 We have has a convergent subsequence. Since bn is 1
... + %t2:k
ai -> 0, k
6
+ 1,
2
it suffices to verify this statement for the case that
SPII -ALGEBRAS
28
where a k ( n ) 1
t
2
0 and a k ( n )
= 0 f o r k s u f f i c i e n t l y large. Since r ( t 2 )
Cl(t2), t h e spectral mapping theorem ( P r o p o s i t i o n P . 3 ) i m p l i e s t h a t
By u s i n g compactness arguments and t h e d i a g o n a l p r o c e s s , we o b t a i n an i n c r e a s i n g sequence I n . ) i n t
+
1
such t h a t f o r e a c h k t h e sequence ( 4 , ( n i ) }
converges. By p a s s i n g t o t h i s subsequence, we m a y suppose t h a t
Bk = exists. Observe t h a t
l i m %(n) n
L e t q be t h e s p e c t r a l idempotent a s s o c i a t e d w i t h t h e s p e c t r a l s e t
and p u t s = t 2 ( e
- q).
S i n c e r ( s ) < 1 , we know t h a t IIsnII * 0 f o r n *
+o.
T h i s i m p l i e s t h a t t h e sequence
converges i n t h e norm of B. L e t d be i t s sum. Then t h e f o r e g o i n g shows t h a t
Now bn
= b
n
(e
-
d = l i m bn(e n
- q).
Hence i n o r d e r t o prove t h a t
q ) + bnq f o r each n i n 2'.
{bn) h a s u convergent subsequence, it s u f f i c e s t o show t h a t {b q] h a s a n convergent subsequence. But (bnq] i s a bounded sequence i n t h e c l o s e d s u b a l g e b r a of B g e n e r a t e d by t q. From t h e s p e c t r a l p r o p e r t i e s o f t 2 i t 2 f o l l o w s t h a t t h i s a l g e b r a i s f i n i t e d i mensiona l ( c f . , P r o p o s i t i o n P.8).
So Ib q ) h a s a convergent subsequence, and t h e proof is complete. n Let t b e a non-zero element i n B s a t i s f y i n g t h e c o n d i t i o n s of t h e p r e v i o u s theorem and suppose t h a t A(t)
- A(t)
U(t)
is an i n f i n i t e set. Then Bo =
is not f i n i t e d i m c n s i o n d . For suppose t h a t Bo is f i n i t e
dimensional. S i n c e t
n
t
+, t h i n
Bo for each n i n Z
inplies t h e existence
of a non-zero polynomial q such t h a t q ( t ) = 0. But t h e n U ( t ) m u s t b e
LOCALLY COMPACT SEMI-ALGEBRAS
29
finite (by Proposition P.31, contradicting our hypothesis. Using Theorem 1.1, it is easy to construct examples of monothetic locally compact semi-algebras. We mention EXAMPLE.
Let T be
a
otle
particular one.
compact linear operator on a complex Banach space
E, and suppose that T has a positive eigenvalue. Then the monothetic semialgebra generated by T in the Banach algebra L(E) of all bounded linear operators on E is locally compact. A semi-algebra A in B is called s t n ' c t i f
A
n
(-A)
f01,
i.e., if -x and x in A implies x = C. The next theorem shovs how one may construct strict locally compact monothetic semi-algebras. The result is formulated in terns of the set Pem(t)
{ A c u(t):
1x1
= r(tI1.
1.2 THEOREM. Let t be a non-zero element of B such that Pens( t) is a s e t of poles of t. Further suppose that r(t) is a pole of t of rnarimal order i n Pem(t). Then A(t) is s t r i c t and locally compact.
PROOF. The proof consists of two parts. First of all we consider the case r(t) > 0. The local compactness of A(t) follovs immediately from Theorem 1.1 by taking a l = 0 and 02 = U(t). 1.
Let s = Bt for some 8
0. Then A(s) = A(t), r(s) = Br(t), and s and t have
similar spectral properties. Hence in order to prove that A(t) is strict, we may suppose without loss of generality that r(t) = 1. Then 1 is a pole of t of order no, s a y . Let po be the spectral idempotent associated with po. Then (by Proposition P.7) (e t)""l the spectral set { l } . Put xo
-
xo # 0 , txo = x0' n Let b(t) = alt + + a t with ai 2 0 (i = 1 n mapping theorem (Proposition P.3) implies that
...
,...,n).
The spectral
r(b(t)) = b(1). Further we have b( t )xo = b( 1 )xo = r(b( t )xo. Take s in A(t) n (-A(t)).
We want to show that s
-
0 . Since
s
c
A(t),
SEMI-ALGEBRAS
30
t h e r e exists a sequence {bn) i n A ( t ) such t h a t s = l i m bn and bn = cri(n)t i ( n = 1,2, . . . I
F=,
with a i ( n ) 1 . 0 and ui(n) = 0 f o r i s u f f i c i e n t l y l a r g e . From what we proved
i n t h e previous paragraph it follows t h a t , By t h e continuity of t h e s p e c t r a l radius on commutative
f o r n = 1,2,.,.
s e t s (Proposition P . 2 ) , t h e last equality implies t h a t
sx0 = r ( s ) x o . Since
-6
(1)
a l s o belongs t o A ( t ) , we can repeat t h e argument t o show t h a t -sx 0 = r(-s)xo.
(2)
zzl
But r(-s) = r ( s ) . Hence ( 1 ) and ( 2 ) together imply t h a t
lim n
r(8)
cri(n) = l i m r ( b n ) = r ( s ) = 0 . n
= 0 , and so (3)
From our hypothesis it follows t h a t t h e peripheral spectrum of t i s a s p e c t r a l set of t . Let p be t h e corresponding s p e c t r a l idempotent. Then r{t(e
- p)) <
1 , and so
tn(e
- p ) = { t ( e - p))"
-+
o
* -1.
(n
I n p a r t i c u l a r t h i s implies t h a t t h e sequence { t n ( e
- p))
i s bounded i n B,
and so by (3)
That i s , s ( e
-
p) = l i m bn(e
- p) = 0.
Hence in order to prove t h a t s = 0 ,
it is s u f f i c i e n t t o show t h a t s p = 0. Let Peru(t) =
lAil
Then
r 1.
-
= 1 , A 1. i s a pole of t of order ni say,and ni < no ( i = 0,1,2,...
..., r ) . Let p.
{Ai}.
{A~=I,A~,...,A
Then
1
be t h e s p e c t r a l idempotent associated with t h e s p e c t r a l s e t
p = Po + p , +
... + pr.
Hence it s u f f i c e s t o show t h a t
Put oro(") = 0 f o r each n
for
c
0,1,...
,r and j = 0,1,...
.
and each n i n 2+ Observe t h a t f o r f i x e d
LOCALLY COMPACT SEMI-ALGEBRAS
31
c = 0. Consider t h e complex n and 1and f o r j s u f f i c i e n t l y l a r g e @.(XI) J
polynomi a1
a. =
for
..,I and each n i n Z+ . A simple computation
0,1,.
shows t h a t
I n particular
,. ..,r and each n i n . Note t h a t
f o r & = 0,l
,r
f o r e = 0,1,... Since
xo
and each n i n
and j = 0,1,2,...
= 1 i s a p o l e of t of o r d e r n Po, ( t
-
Xo)po,
a r e l i n e a r l y independent. Further
Now s p
0
(4)
+ i3
0'
(5)
+ Z .
t h e elements
... , ( t - x F 0 - l Po n ( t - xo) Op0 = 0 , and t h u s by
(6)
0
(4)
= l i m bnpo. Hence
where
-
..,n
-1).
But t h e elements ( 6 ) a r e l i n e a r l y
Note t h a t ( 5 ) implies t h a t BT)(-)
Also
> 0 f o r j = 0,1,.
J
-13
-1.
0
i n A ( t ) . So r e p e a t i n g t h e argument we see t h a t
f o r some yo,
0 ( j = O,l,...,n
J
0
independent, t h e r e f o r e ( 7 ) and ( 8 ) t o g e t h e r imply t h a t BT)(-) J - 1 , and hence by (5)
j = O,l,...,n
0
e
l i m Bj(n) = for
(t
e= -
n
O,l,...,r
Since h he)"'pL
e
and j = 0,1,...
o
(9)
,n -1. 0
i s a pole of t of order ne with ne'n
= 0. Hence (4) implies t h a t
= 0 for
0
, we
have
SEMI-ALGEBRAS
32
t1
for
. .,r and each n i n Z+ , But then w e can use ( 9 ) t o show t h a t
0,1,.
spa. = l i m bnpr = 0 n
(1 = O , l ,
..., r ) .
This completes t h e proof of p a r t 1 .
u ( t ) = {O), and t h e l o c a l
2 . Next we consider t h e case r ( t ) = 0. Then
compactness of A ( t ) follows immediately from Theorem 1 . 1 by taking Ul
=
U ( t ) and U2
= 0. Observe t h a t t h e assumption r ( t ) = 0 together with
the hypotheses of t h e theorem implies t h a t t i s nilpotent. Let no be t h e order of nilpotence. If no = 1 , then t = 0 , contradicting t h e hypotheses of t h e theorem
Thus no
2 2.
Take s i n A ( t ) n ( - A ( t ) ) . We want t o show t h a t s = 0. Let t h e sequence be as i n p a r t 1 . Then
{bn) i n A ( t
...). no- 1 Using t h e l i n e a r independence of t h e elements t , t 2 , .. . ,t we see t h a t bn =
lyzl-1
ai(n)t
s
where
l i m ai(n)
Ui(m)
=
-
> 0 (i = 1,.
i-
( n = 1,2,
Ql-1
0 for i = 1
repeat t h e argument t o show t h a t
f o r some 6
i
i
ai(4t
,. ..,n
-1.
0
,
(10)
Also -s i n A ( t ) . So w e can
..,n ) . Since t h e elements t , t2,...,tn -1 -1
0
are
l i n e a r l y independent, ( 1 0 ) and ( l l ) together imply t h a t ai(=) = 0 f o r
i
l,.,,,n -1. Thus s = 0. This completes t h e proof. 0
One of our aims is t o get converses of t h e preceding two theorems. For Theorem 1 . 1 t h i s w i l l be t h e main t o p i c of Chapter 111. The results of Chapter 111 a r e based on t h e existence of idempotents i n l o c a l l y compact semi-algebras, a subject we w i l l be dealing with i n t h e next section.
2. IDEMPOTENTS
I n t h i s section B denotes a complex Banach algebra with u n i t element e. L e t A be a semi-algebra i n B. A non-empty subset J of A i s c a l l e d a r i g h t
ideal of A i f ( i ) J i s a semi-algebra,
( i i ) x c A, a E J implies
J. Similarly one defines a l e f t i d e a t . The set J i s s a i d t o be a two-sided BX f
LOCALLY COMPACT SEMI-ALGEBRAS
33
ideat i f J i s both a l e f t i d e a l and a r i g h t i d e a l . An i d e a l ( l e f t , r i g h t or two-sided) J of A i s c a l l e d a closed i d e a l i f J i s a closed subset of A with respect t o t h e r e l a t i v e topology induced i n A by t h e norm topology of B. Note t h a t closed i d e a l s i n a l o c a l l y compact semi-algebra are closed
subsets of B. A closed r i g h t i d e a l J of A i s c a l l e d a minimal closed right ideal i f J
# ( 0 ) and t h e only closed r i g h t i d e a l s of
A contained i n J are ( 0 ) and J.
Similar d e f i n i t i o n s apply f o r minimal closed l e f t o r two-sided i d e a l s .
I n a locally compact semi-algebra each non-aem closed right ideal contains a minimal closed r i g h t ideal. A similar statement holds for $ e f t and two-sided ideals. 2.1 LEMMA.
PROOF.
L e t J be a non-zero closed r i g h t i d e a l of t h e l o c a l l y compact
semi-algebra A. Further l e t W be t h e family of a l l non-zero closed i d e a l s of A contained i n J. The s e t W i s non-empty, and W i s p a r t i a l l y ordered by the inclusion r e l a t i o n
c.
The ordering on W i s inductive. To see t h i s , l e t
V be a non-empty l i n e a r l y ordered subset of W. Put
v,
= {I n ( X E B:
llxll = 1 ) :
I
E
v).
Then V 1 i s a family of compact sets and V 1 has t h e f i n i t e i n t e r s e c t i o n property. Hence n V 1 # 0 and thus Jo = n V contains non-zero elements. Note t h a t Jo i s a closed r i g h t i d e a l of A contained i n J. Hence Jo i s a lowerbound f o r V i n W. So W i s inductively ordered, and w e can apply Zorn’s lemma t o show t h e existence of a minimal closed r i g h t i d e a l contained i n J. The “minimum condition” on t h e closed i d e a l s proved above i s t h e main t o o l i n t h e s t r u c t u r e theory of l o c a l l y compact semi-algebras. I n t h i s s e c t i o n we develop t h i s theory a s f a r as i s necessary t o g e t s u i t a b l e conditions ensuring t h e existence of idempotents. L e t D be a non-empty subset of t h e semi-algebra A. The right annihita-
t o r D of D i s t h e set of a l l x i n A such t h a t r
D~ = {X E A:
BX
=
o
(a
BX E
= 0 f o r each a i n D. Thus D)).
A l e f t annihilator i s defined s i m i l a r l y .
Observe t h a t Dr i s a closed r i g h t i d e a l of A. If D i t s e l f i s a r i g h t i d e a l , then it i s e a s i l y seen t h a t D
r
i s a closed two-sided i d e a l .
34
SEMI-ALGEBRAS
Let E be a closed subset of the locally compact semialgebra A and SUppose that aE c E ( a 2 0 ) . Let a be an element i n A such that 2.2 LEMMA.
{a)r n E = (0).
Then aE
(1)
Closed.
$8
PROOF. Let y = l i m axn, where {xn ) i s a sequence i n E. F i r s t l y , suppose t h a t t h e sequence Ex 1 i s bounded. The l o c a l compactness of A n implies t h e existence of a convergent subsequence { x ) with l i m i t xo, say. "i Then y = axo, and xo E E , since E i s closed. Hence y E aE, which i s t h e desired r e s u l t .
II~~I~I
Next suppose (x-1 t o be unbounded. Then t h e r e e x i s t s a subsequence
2
{xn I such t h a t i
. . . put
i f o r i = i,2,.
x.I = IIxn l l - l x 1 i "i
( i = 1,2,
...1.
Note t h a t l i m ax( = 0. The hypothesis on E implies t h a t xf E E f o r each i . Since
IIxiII
-
= 1 , we can repeat t h e argument of t h e first p a r t of t h e proof
t o show t h a t { x i ) has a limit point xd i n E and ax
I
0
0. By ( l ) , x:
= 0.
However, t h i s contradicts t h e f a c t t h a t IIx
i s not unbounded.
An idempotent p i n A is c a l l e d a r i g h t minimal idempatent of A i f PA
i s a minimal closed r i g h t i d e a l . Note t h a t minimal idempotents are non-zero. 2.3 PROPOSITION.
Let A be a locally compact semi-algebra, and l e t M be a minimat c b s e d e h t idea1 i n A. Suppose that p is a no??-zero idempotent i n M. Then pm = m
(m
E
MI.
In p a r t h Z a r pA = M, and thus p i s a right minimal idempotent. PROOF.
Put J = {m
E
M: m
- pm c MI.
Then J i s a closed r i g h t i d e a l
contained i n M. Since 0 # p E J and M i s m i n i m a l , we have J = M. Each element m
- pm
E
{pjr. Hence t h e foregoing implies t h a t
m
- pm
E
{pJr n M
(m
E
M).
The set {p), n M i s a closed r i g h t i d e a l contained i n M. Since p
4 {PI
r'
LOCALLY COMPACT SEMT-ALGEBRAS
{ p l r n M # M. Hence {p), n M = ( 0 ) , and thus m = pm f o r each m
35 E
M. The
remaining a s s e r t i o n s are t r i v i a l consequences of t h i s r e s u l t . 2.4 PROPOSITION.
Let A be a l o c a l l y compact semi-algebra, and l e t M be a minimal closed r i g h t ideal i n A. Take a i n A and suppose that aM # (0). Then eM is a minimal closed r i g h t i d e a l . The set {a)
PROOF.
r n M i s a closed r i g h t i d e a l of A contained i n M. From our hypotheses it follows t h a t (a), n M # M. Since lul i s minimal, t h i s implies t h a t {a),
n M =
(0).
So we can apply Lemma 2.2 t o show t h a t aM i s closed.
Clearly aM i s a r i g h t i d e a l . It remains t o show t h a t aM i s minimal. L e t I be a closed r i g h t i d e a l contained i n aM, and suppose t h a t I
L e t J = {m
t
# aM.
M: am E I ) . The s e t J i s a closed r i g h t i d e a l contained i n M
and J # M. Hence J = ( 0 ) , and thus I = ( 0 ) . This shows t h a t aM i s a minimal closed r i g h t i d e a l .
Let A be a l o c a l l y compact semi-a1gebm, and l e t M be a m i n k 1 closed r i g h t ideal i n A. Suppose that M2 # (0). Then M contains a non-xem, idempotent. 2.5 THEOR&.
PROOF.
The proof c o n s i s t s of f o u r p a r t s .
1 . From our hypotheses follows t h e existence of an element t i n M such
t h a t t M # ( 0 ) . L e t J = {t), n M. Then 3 is a closed r i g h t i d e a l contained i n M and J # M. Since M i s minimal t h i s implies t h a t J = (0). Hence tx # 0
(0
# x
E
M).
2. Assume t h a t A i s commutative. From ( 2 ) it follows t h a t xM
(2)
# (0)f o r
each non-zero x i n M, and hence we can apply Proposition 2.4 t o show t h a t xM i s a closed r i g h t i d e a l . Further XM c M ( x E M ) . So
xM=M
(OfxcM).
But t h i s implies t h a t M \ 10) i s a group with respect t o multiplication. The unit element of t h i s group i s a non-zero idempotent contained i n M. This completes t h e proof f o r t h e commutative case.
3. Next we r e t u r n t o t h e general case. Let t be as i n p a r t 1. Consider A ( t ) , t h e monothetic semi-algebra generated by t . Since A ( t ) c M , (2)
36
SEMI-ALGEBRAS
implies t h a t t s # 0 f o r each non-zero s i n A ( t ) . But A ( t ) i s commutative. Thus st # 0 f o r 0
# s
E
A ( t ) . Hence
s M # (0)
(0
# s
A(t)).
E
Repeating t h e arguments o f p a r t 1 f o r ' s i n s t e a d o f t we see t h a t sx # 0
(0
#
x
E
M).
I n p a r t i c u l a r s 1 s 2 # 0 f o r each p a i r s 1 and s2 of non-zero elements i n A ( t 1.
4. Observe t h a t A ( t ) i s a commutative l o c a l l y compact semi-algebra. According t o Lemma 2.1, A ( t ) contains a minimal closed i d e a l N , say. Since A ( t ) has no d i v i s o r s of zero ( s e e t h e conclusion of p a r t 31, w e have N2 # ( 0 ) . But t h e n , by t h e result of p a r t 2 , N c o n t a i n s a non-zero idempotent p.
Since N c A ( t ) c M,
p i s a non-zero idempotent i n M. An element p i n a semi-algebra A i s c a l l e d a unit ei!ement of A i f p # 0 and pa = ap = a
(a
E
A).
A semi-algebra A i s c a l l e d a division semi-algebra i f A has a u n i t element
p and each non-zero element x of A has an inverse y i n A , i . e . xy = yx = p.. Equivalently, A i s a d i v i s i o n semi-algebra i f A* = A \ (0)
i s a group w i t h r e s p e c t t o m u l t i p l i c a t i o n . 2.6 THEOREM.
Let p be a r i g h t minimal idempotent i n the localtg compcct semi-algebra A. Then pAp is a locaZty compact division semi-atgebm. PROOF.
Let D = PAP.
Then D is c l e a r l y a semi-algebra w i t h u n i t
element p. Further D i s a closed subset o f A , s i n c e
D = {x
E
A: x = px = xp}.
The l o c a l compactness of A implies t h a t D is l o c a l l y compact. It remains t o prove t h a t D* = D \ EO) i s a group with r e s p e c t t o
m u l t i p l i c a t i o n . Let pap be i n D*. We s h a l l prove t h a t t h e equation
37
LOCALLY COMPACT SFMI-ALGEBRAS
To show t h i s , it s u f f i c e s t o prove t h a t t h e equation
has a solution i n D*.
i s solvable i n PA.
From our hypothesis it follows t h a t M = pA i s a minimal closed r i g h t i d e a l of A. Since p2 = p I 0
# pap
(pa)M.
E
So w e can apply Proposition 2 . k t o show t h a t (pa)M i s a closed r i g h t i d e a l .
But (pa)M
c
(pA)M = M2 c M.
Since M i s minimal, t h i s implies (pa)M = M. Hence equation (4) has a solution i n M = PA. The f a c t t h a t equation ( 3 ) i s solvable in D* proves t h a t each element i n D* has a r i g h t inverse i n D*. Using a standard argument from elementary group theory, one can show t h a t t h i s implies t h a t D* i s closed under multiplication. Hence it follows t h a t D
2.7 THEOREM. element p. Then
is a group with u n i t element p.
Let A be a strict closed d i v i s i o n semi-algebra w i t h unit
+
A=IRp. 0
PROOF.
Let u
E
A. F i r s t of a l l we show t h a t hp
-u
E
A for
> llull.
Consider t h e s e r i e s
with uo = p. For 1 >
1 1,1 1
t h i s s e r i e s converges i n B. Let bh denote i t s
sum. Since t h e p a r t i a l sums of t h e s e r i e s belong t o A and A i s closed, b
x
E
A. Further we have bh(hp
- u) =
(hp
x p - u = bh' Here x-'
-
u ) b h = p. Hence E
A.
denotes t h e inverse of x i n A.
L e t p = i n f { A : hp closed, pp
-u
L
-u
E
A). Since A i s s t r i c t , p
A. W e s h a l l prove t h a t
y = pp
-u
= 0.
2 0.
Since A i s
38
SEMI-ALGEBRAS
Suppose not. Then, by t h e argument of t h e f i r s t paragraph, vp
- y-1
E
A for
v s u f f i c i e n t l y l a r g e and p o s i t i v e . But then
f o r E > 0 s u f f i c i e n t l y s m a l l . However t h i s c o n t r a d i c t s t h e d e f i n i t i o n of p . Hence y = 0 and u = pp. This shows t h a t A
+
c
+
mop.
Since p
E A
t h e reverse
i n c l u s i o n holds t r i v i a l l y . Thus A = IR p. 0
3. SIMPLE LOCALLY COMPACT SEMI-ALGEBRAS A semi-algebra A i s s a i d t o be simple i f ( 0 ) and A a r e t h e only two-
sided i d e a l s of A. Clearly a d i v i s i o n semi-algebra i s simple. The nxn matrices with non-negative e n t r i e s provide a l e s s t r i v i a l example. Let M ( C ) be t h e a l g e b r a of a l l nxn matrices with complex e n t r i e s n endowed with some a l g e b r a norm. With such a norm'Mn(C) i s a Banach algebra.
+
Denote by M n ( I R o )
t h e semi-algebra of a l l matrices belonging t o M n ( C ) t h a t
have non-negative e n t r i e s . This semi-algebra i s l o c a l l y compact, s t r i c t ,
+
simple and it has a u n i t element. The next theorem shows t h a t Mn(lRo)
is
c h a r a c t e r i z e d by t h e s e f o u r p r o p e r t i e s . 3.1 THEOREM. Let A be a 1ocaZZy compact, s t r i c t , simple semi-algebra w i t h a unit element. Then there e x i s t s a p o s i t i v e integer n such thut A is + isomorphic to the semi-algebra Mn( lRo )
.
The proof goes i n a number of s t e p s . I n t h e sequel A i s a l o c a l l y
compact, s t r i c t , simple semi-algebra with u n i t element 1. Since A # ( 0 ) ,
w e know t h a t 1 # 0. Given non-empty s u b s e t s B and D of A, w e denote by BD t h e set of a l l f i n i t e sums bld, + with bi
3.2
E
B and d.
1
E
. . a
+ bndn
D.
Let M be a nun-zero r i g h t ideal i n A. Then M2 #
PROOF.
(0).
Suppose M2 = ( 0 ) . Then (AM)(AM)
=
AMz = ( 0 )
Note t h a t AM i s a two-sided i d e a l , and AM # ( 0 ) s i n c e 1
(1) E
A. But A i s
LOCALLY COMPACT SEMI-ALGEBRAS simple, t h e r e f o r e AM = A. Then ( 1 ) implies t h a t A' fact that 0 # 1
A*. Hence M2
6
#
39
= ( 0 ), contradicting t h e
(0).
Since A i s l o c a l l y compact, Lemma 2.1 implies t h e e x i s t e n c e of a minimal closed r i g h t i d e a l i n A. By what we have j u s t proved, each minimal closed r i g h t i d e a l i n A s a t i s f i e s t h e conditions of Theorem 2.5. Hence there exist
r i g h t minimal idempotents i n A ( s e e Proposition 2 . 3 ) . Let
denote t h e s e t of a l l r i g h t minimal idempotents i n A . Then 3.3.
TI # 0.
TIA = A.
PROOF.
Observe t h a t n A i s a non-zero r i g h t i d e a l of A. Since A is
simple, it s u f f i c e s t o show t h a t RA i s a l e f t i d e a l , t h a t i s (2)
A ( n A ) c TIA
and a i n A. If ap = 0 , t h e n apx = 0
Take p i n
#
Suppose ap
E
JfA f o r each x i n A.
0. Then ( a p ) ( p A ) # ( O ) , and we can apply Proposition 2.4 t o
show t h a t (ap)(pA) i s a minimal closed r i g h t i d e a l i n A. Each minimal closed r i g h t i d e a l i n A s a t i s f i e s t h e conditions of Theorem 2.5. Hence there exists q
E
TI
such t h a t (see Proposition 2.3) (ap)(pA) = qA.
But then ap = qb f o r some b i n A , and t h u s apx = q ( b x ) E n A f o r each x i n A. Hence ( 2 ) holds.
For p and q i n
3.4. PROOF.
n,
the s e t pAq # ( 0 ) .
Suppose pAq = ( 0 ) . Then q
E
(PA),.
The s e t (PA),
i s a two-
s i d e d i d e a l containing a non-zero element, namely q . Therefore (@), s i n c e A i s simple. However, p Thus pAq
p
4 (PA),.
s
A
Contradiction.
(0).
Given p and q i n II, there e x i s t s a rwn-zero u i n A such that
3.5.
pAq = IR+.
If p = q,
then we may take u t o be p.
PROOF.
Take v = qap # 0 i n qAp. Then v( PA)
( 0 ) and t h u s , by
Proposition 2 . 4 , t h e set v(pA) i s a minimal closed r i g h t i d e a l of A. From v(pA) c qA we s e e t h a t v(pA) = qA, and t h e r e f o r e
SEMI-ALGEBRAS
40
v(pAq) = q k . Hence t h e r e e x i s t s u i n pAq such t h a t vu = q. Observe t h a t (pAq)v = pAqap c PAP.
+
According t o Theorems 2.6 and 2.7, pAp = Bop. So
+
+
0
0
pAq = (pAq)vu c IR pu = IR u. On t h e o t h e r hand u
E
+
+
pAq, and t h e r e f o r e IR u c pAq. Hence pAq = IRou. 0
I f p = q , t h e n by Theorems 2.6 and 2.7 we may t a k e u t o be p. 3.6.
There e x h t s u 1 ,u2,. 1 = u1
PROOF.
i n A ( i = 1, 2 , . .
... + un,
+ u2
Because 1
E A
.. ,un in Jl such that
and
. ,n) such t h a t
nA
u.u = i j
o
( i # j).
= A ( s e e 3 . 3 ) , t h e r e e x i s t ei i n
1 = e a + e a +...+ 1 1 2 2
and a
e a . n n
i
(3)
We may suppose t h a t t h e r e p r e s e n t a t i o n ( 3 ) of 1 is chosen so t h a t n i s as
small a s p o s s i b l e . Postmultiplying ( 3 ) by e l g i v e s el = e a e 1 1 1 By 3.5 we have e l a e l = he (I
Formula
(4) implies
1
... + enan e 1 '
+ e2a2el +
with h
2
0. Hence
- h ) e l = e 2a2e1 + ... + en ane 1 '
that h
2
1, for, if h
(4)
1 , t h e n we can s u b s t i t u t e f o r
e l i n ( 3 ) and we o b t a i n a r e p r e s e n t a t i o n of t h e u n i t element o f t h e form 1 = e2b2
+
,..
+
say, which c o n t r a d i c t s t h e d e f i n i t i o n of n. Therefore h L 1 . On t h e o t h e r hand, because of t h e s t r i c t n e s s of A , formula
(4) shows
that h > 1 is
impossible. Thus h = 1 and hence e ae
= e l , e2a2el +
... + enan e 1 = 0.
Again using t h e s t r i c t n e s s of A , w e s e e t h a t t h e second p a r t of t h e l a s t formula implies t h a t e . a . e
1 1 1
= 0 f o r i = 2,...,n.
By r e p l a c i n g e l by e j i n
t h e preceding arguments, we o b t a i n e.a.e
= e J J ~j y Let ui = e. a. ( i = 1 , ,n) 1 1
e.a.e
i i j
=
o
( i # j).
... . Then we have proved t h a t
LOCALLY COMPACT SEMI-ALGEBRAS
1 = u
# u. 1
Since 0
... + un ,
+
1
= u? = u.( e . a . ) 1
1
1
1
u u = i j
u? = u. 1
, we
1'
41
o
(i # j ) .
# ( a ) , and s o we can a p p l y
have u.( e i A )
P r o p o s i t i o n 2.4 t o show t h a t u . ( e . A ) i s a minimal c l o s e d r i g h t i d e a l . C l e a r l y , u. 1
3.7.
1
E
1
u . ( e . A ) . But t h e n , by P r o p o s i t i o n 2 . 3 , u. 1
1
E
Jl.
For each i and j we have
(5)
u1 . ~ Ju .= R 0 + e1J' ..
where e . . is some non-zero element of u.Au 1J
j'
1
so that
e i i = u. ( i = 1 ,.
.. , n ) ,
S i n c e u.
E
1J
= e e. e ij jk i k ( i , j ,k = 1
,. . , , n ) ,
( j # k).
e. .e = 0 ij kh PROOF.
The eZements e . . can be chosen
TI, we know from 3.5 t h a t t h e r e e x i s t e l e m e n t s e . .
s a t i s f y i n g ( 5 ) . Moreover, a g a i n a c c o r d i n g t o 3.5, we can choose e .
u. f o r i = 1 ,
... ,n.
Observe t h a t t h e c o n d i t i o n e. .e IJ
kh
1J t o be
( j # k)
= 0
i s automatically s a t i s f i e d since uj\ the e
ii
= 0 f o r j # k . I t r e m a i n s t o choose
so t h a t e . e = ei k ' ij i j jk F o r t h e e l e m e n t s el.i w i t h j # 1 , w e t a k e a r b i t r a r y non-zero e l e m e n t s
o f u,Auj. Then ( 5 ) h o l d s f o r t h e s e e l e m e n t s . Next, l e t j
# 1 and k b e
a r b i t r a r y . The l e f t a n n i h i l a t o r (Au ) i s a two-sided i d e a l n o t c o n t a i n i n g k e S i n c e A i s s i m p l e , t h i s i m p l i e s t h a t (A\)L = ( 0 ) . Therefore
yc.
(0)
# e l jAuk = e , jujA\
c ulA\
and hence eljujA\
= ulA\.
T h i s shows t h a t t h e e q u a t i o n eljX =
(6)
lk
h a s one ( a n d no more t h a n o n e ) s o l u t i o n i n u.A \, e j k s a y . We have now
Observe t h a t f o r j = k 2 2 t h e s o l u t i o n jk' 0 and b e l o n g s t o ujA\. Thus ( 5 ) h o l d s .
completed t h e c h o i c e o f t h e e of
( 6 ) is u.. Further e . #
J Jk According t o ( 6 ) . w e have e l j e j k = e l k f o r j
we obtain eljejk = elk
( j ,k = 1
#
1. Since el l e l k = elk'
,. . . , n ) .
SEMI-ALGEBRAS
42
-
Take a r b i t r a r y i , j and k. From eijejk E u.A\ 1
+
Doeik,
= hik. But then jk e l k = e l i ( e i j e j k ) = Xelieik
w e conclude t h a t e. .e IJ
p
Xelk.
This implies t h a t X = 1, and hence eijejk = eik f o r all i , j and k . We a r e now able t o prove Theorem 3.1. Take x i n A. Then x = 1.X.l The map x
+
(hij)y,j=l
=
is t h e desired isomorphism.
NOTES ON CHAPTER I1 1 . I n [ 4 1 Bonsall and Tomiuk have proved t h a t t h e monothetic semi-
algebra generated by a compact l i n e a r operator T which has t h e additional property t h a t 0 < r(T)
B
u(T),
i s l o c a l l y compact. This r e s u l t i s t h e f i r s t version of Theorem 1 . 1 . A second version of the theorem appears i n C191, where it waa shown t h a t t h e condition t h a t T i s compact can be replaced by t h e requirement t h a t the peripheral spectrum of T c o n s i s t s of poles of T. 2. It i s easy t o construct an example of a compact operator T with
r ( T ) = 0 which generates a semi-algebra which is not l o c a l l y compact. Let
el be
t h e Banach space of absolutely summable sequences x = {xn) of
complex numbers, and l e t T on 1, be defined by (Tx), =
n+ n+ 1 X
1
( n = 1,2,...)
Then T i s compact and quasi-nilpotent ( i . e . , r ( T ) l o c a l l y compact ( s e e C191 f o r d e t a i l s ) .
-
. O ) , but A ( T ) i s not
3. Each nilpotent operator generates a semi-algebra which is l o c a l l y compact. Whether a quasi-nilpotent, non-nilpotent l i n e a r operator can generate a l o c a l l y compact semi-algebra i s an open problem.
43
LOCALLY COMPACT SEMI-ALGEBRAS
4. The f i r s t example o f an i n f i n i t e dimensional l o c a l l y compact semia l g e b r a i s given by Bonsall i n C21. It goes as follows. L e t A b e t h e s e t of
real numbers c o n s i s t i n g of t h e closed u n i t i n t e r v a l [0,11 t o g e t h e r w i t h t h e
r e a l number 2 , and l e t A be given t h e r e l a t i v e topology induced from t h e u s u a l topology on t h e r e a l l i n e . Denote by A' t h e c l a s s of real-valued f u n c t i o n s defined on t h e closed i n t e r v a l [0,21 t h a t are continuous, nonn e g a t i v e , i n c r e a s i n g and convex t h e r e . F i n a l l y , l e t A denote t h e c l a s s of f u n c t i o n s on A t h a t a r e t h e r e s t r i c t i o n s t o A of f u n c t i o n s i n A ' . Then A i s
a l o c a l l y compact semi-algebra not contained i n a f i n i t e dimensional a l g e b r a ( s e e [21 f o r d e t a i l s ) .
5. Theorem 1.2 i s r e l a t e d t o work of L. Elsner. I n c111 E l s n e r proved t h e following theorem. L e t T be a compact l i n e a r o p e r a t o r on a complex Banach space E. Suppose t h a t r ( T ) > 0 and t h a t r ( T ) i s a p o l e of T of maximal o r d e r i n t h e p e r i p h e r a l spectrum of T. Then t h e r e e x i s t s a closed and t o t a l cone ( s e e s e c t i o n IV.2 f o r t h e d e f i n i t i o n s of t h e s e n o t i o n s )
E such t h a t TK c
K.
The f a c t t h a t
K is a
K
in
closed and t o t a l T-invariant cone
implies t h a t A ( T ) i s s t r i c t ( s e e Chapter IV f o r more r e s u l t s of t h i s t y p e ) . Hence f o r o p e r a t o r s T of t h i s kind t h e f i r s t p a r t of Theorem 1.2 follows from E l s n e r ' s theorem ( c f . Theorem 8 i n C171). Whether o r not E l s n e r ' s theorem holds f o r o p e r a t o r s T s a t i s f y i n g t h e c o n d i t i o n s of Theorem 1.2 i s a n unsolved problem.