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.