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A COMMUTATIVE SEMISIMPLE ANNIHILATOR
BANACH ALGEBRA WHICH IS NOT DUAL
BY B. E. JOHNSON
Communicated by E. Hewitt, December 7, 1966
Barnes [l] has constructed an example of a commutative semisimple normed annihilator algebra which is not a dual algebra. His
example is not complete and when completed acquires a nonzero
radical. In this paper we construct an example which is complete.
The theory of annihilator algebras is developed e.g. in [2].
We putai=(l
+ (l+i) 1i2)~2 for i^l and denote by A0 the algebra
of doubly infinite sequences a with a; = 0 for all b u t a finite number
of values of i, with coordinatewise addition and multiplication. We
define a norm on A 0 by
Ml = 3 ( 2 : | a n | 2 )
2
+ 3sUP
&n&n
"
/ .j &j
This is easily seen to be a linear space norm on Ao and we have that
(EI«AI,Y%(EI«.I,Y/YEI«W|T'
«
**MIIWI;
(ii)
if n > 0,
0
*H ^
j——n
à I On I On — 0 + 1) i||fl!
so that
a n l a n ^ K l + Cn+l) 1 7 2 )!^!!
and
An^nOn
(iii)
0
|
â
3«n(l +
( « + ! )
m\\*b
A1/2/
I / o
) | | a | | ||&||
°
A1
£«**,*( I) k l ) ( E N )
j——n
I
\
ƒ=»—w
/
filial lift
407
\
j——n
/
408
B. E. JOHNSON
[May
The submultiplicative property of ||-|| follows easily from (i), (ii)
and (iii).
Consider now the space Z2( — °°, 0)©c(l, °o)> which we consider
as a space of doubly infinite sequences, with norm as the sum of the
k norm and the sup norm. For a £ ^ 4 0 define T0 a by
(T0a)n = 3x'2an
for n^O,
__!
o
= 3anan — 3 ]T) ai
f° r ^ > 0.
To is then a linear isometry; AQ—»Z2( — °°, 0) ©c(l, <*>). The multiplicative linear functional on A 0 are
<£»(a) = 0»
and if the functionals \f/i on k®c are defined by
He)
= a/Vi*
for i ^ 0,
= (ci + (31'2) 2
cy)a</3
for * > 0,
then the ^ are continuous and T*ypi=<l>i. The set {\[/i; i £ Z } is
clearly total on Z 2 0c.
Let now ^4 be the completion of A0; To extends to an isometry
T; A—>h@c, the 4>% extend to multiplicative linear functionals on A
and <t>i=T*\f/i. Since the \pi are total on Z2©c, the <£»• are total on A
and A is a semisimple Banach algebra. Writing di=<t>i(a) for at—A
we can consider the elements of A as doubly infinite sequences and
the two ways in which an element of A 0 becomes a sequence give the
same sequence.
If hi is the sequence in A0 with (ô»-)i = ô»y (the Kronecker symbol)
then aôi = diôi for all a in ^4 so that if / is an ideal in A either 8,-£ J
or 8 t / = {0}. Thus if / is a closed ideal in A with zero annihilator then
all the 8i are in J , A0C.J and J = ^4. Hence ^4 is an annihilator algebra.
The span J0 of the set {5»; i > 0 } is an ideal in ^4 and thus so is its
closure J. The annihilator of J is
i£ = {b: bE A,bi = 0iori>
0}
and the annihilator of X" is
J = {c: c£
A, a = O f o r i ^ 0J.
The norm on J is given by ||c|| = sup|c»-/a»| and since an~o(cLn)
as #—>oo for aÇzJo we have a n = (?(ajw) for a £ J " . Define a sequence xn
from ^40 by
1967]
A BANACH ALGEBRA WHICH IS NOT DUAL
(*n)i = - 1 / 0 * + 1 ) ,
= 0,
409
- n ^ i Û 0,
i < — n,
= f 1 + £ aw J a{,
i> 0.
.7'
Then, since the supremum term is 0,
O 1 / ,, j Xmi
\ *^o
Xni J J
/
â 3 ( E |^,| 2 )
*S0
2
+ 3 ( i : \xni\*)
/
N *â
1 2
= 3(w + l ) " ' + 3 0 + I)" 1 ' 2 ,
so t h a t xn converges to a limit y m A. We have
j t - = lim Xni = 0,
if i ^ 0,
n
= ai,
if i > 0.
Clearly yÇ:J but yi = ai9^0 (aï) so t h a t y » $ 7 , the ideal / is not an
annihilator and A is not a dual algebra.
The question of the existence of simple annihilator Banach algebras
which are not dual remains open!
REFERENCES
1. B. A. Barnes, An annihilator algebra which is not dual, Bull, Amer. Math. Soc.
71 (1965), 573-576.
2. C. E. Rickart, General theory of Banach algebras. Van Nostrand, New York,
1960.
THE UNIVERSITY, NEWCASTLE UPON TYNE, ENGLAND.