10 W. G. BADE, H. G. DALES, Z. A. LYKOVA
and so b b2 and
r 3 = (p - b)3 =p3- 3p2b + 3pb2 -bs=p-b = r.
Since r 2 rad A, there exists s G rad A with r2 + s = r 2 s . But now
r = r r(r2 + s r2s) = (r r 3 ) (r r 3 )s = 0,
and p = b £ B.
Now let (^2, || ||2) be the standard Banach space with pointwise multiplication, and
set 21 = I2 © Cr as a linear space, with the product
(a + ar)(b + /3r) = ab.
Then 21 is a commutative algebra and rad 21 = {0} © Cr, so that dim rad 21 = 1. Let A
be a linear functional on £2 such that A | £x is the functional
oo
(a
n
) i—• y ^ Q n ,
n =l
and set
|||a + zr||| - max{||a||2 , |A(a) - z\} (a e£2, zeC).
It is easily checked that (21, ||| |||) is a Banach algebra, and that 21 splits algebraically,
with the decomposition 21 = £2 © Cr. Now assume that 21 = 03 © Cr for a closed
subalgebra 03 of (21, ||| |||). By the above remark, coo is contained in 03. However it is
easily seen that coo is ||| |||-dense in 21, and so 03 = 21, a contradiction. Thus ^ does
not split strongly.
There is one case in which Question 1 can be solved in a trivial way, and we first
dispose of this case; it leads to some easy reductions of the general problem.
1.4. PROPOSITION. Let XX^5 I) be an extension of a Banach algebra A.
(i) Suppose that I contains a non-zero idempotent p such that I = pi + Ip. Then
][](2l; /) splits strongly.
(ii) Suppose that I is semisimple and finite-dimensional. Then ^(21; /) splits
strongly.
PROOF: (i) We have
I = p2t + 2lp = pQL(e - p) + (e - p)QLp + p$lp.
Set
B=(e- p)2l(e - p).
Then B is a closed subalgebra of 21 and 21 = B + / . Now take a G # D / . Then there
exist b E 21 and c,d e I with
a = (e p)6(e p) = pc + dp,
and so
a = (e p)a(e p) = (e p)pc(e p) + (e p)dp(e p) = 0.
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