SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS

11

Thus B D J = 0, and so 21 = B 0 J and £(21; I) splits strongly.

(ii) The hypotheses imply that / has a non-zero identity. D

Let ^2 — X X ^ ^ ) D e a n extension of a Banach algebra A such that I contains a

non-zero idempotent p with I = pi + Ip + Ipl, so that / is the ideal generated by p.

Certain later proofs would be shorter if it were the case that this implied that Y2 splits

algebraically. However this is not necessarily the case: we have an example of a unital,

finite-dimensional algebra 21 and an idempotent p such that the sequence

0 —• 2lp2l — 21 —• 2l/2lp2l —• 0

does not split algebraically.

In fact the case where / is semisimple is only considered in a minor way in this memoir.

For the theory of semidirect products in the case where 21 (and hence I) is a semisimple,

commutative Banach algebra, see the thesis [Bern]. Indeed, in this interesting thesis, the

author begins with a careful discussion of a more formal definition of semidirect products

in various categories, including that of Banach algebras and continuous homomorphisms;

the notion of semidirect product given in the thesis coincides with our notion of a short

exact sequence which splits strongly.

Let X X ^ I) b e a n extension of a Banach algebra A, and let J be a closed ideal of 21

with J C I. Then we can consider an extension

]T(2l/ J; I/J) : 0 —• 1/ J ^ + 21/ J ^ A — 0,

where

Lj(x + J) = i(x) + J (X £ I)

and

7rj(a + J) = 7r(a) (a e A).

It is immediate to check that 5Z(2l/J; I/J) is an extension of A, and that it is admis-

sible (respectively, commutative) in the case where X X ^ -0 *s admissible (respectively,

commutative). Suppose that 6 : A — 21 is a splitting homomorphism for XX^5 -0 • Then

clearly, the map a i—• 0(a) + J , ;4 — 21/J, is a splitting homomorphism for $Z(2l/J; / / J ) .

The following result will be very useful in inductive proofs, for it will allow us to reduce a

given situation to a more elementary one.

1.5. PROPOSITION. Let ^2 — X X ^ I) be a [commutative] extension of a Banach algebra

A, and let J be a closed ideal of 21 with J c l .

(i) Suppose that every [commutative] extension of A by J splits strongly and that

the extension XX^/^5 11 J) splits strongly. Then J2 splits strongly

(ii) Suppose that every [commutative] extension of A by J splits algebraically and

that the extension 53(21/J; I/J) splits strongly Then J2 splits algebraically

PROOF: (i) The [commutative] extension

^ ( 2 1 / J ; I/J) : 0 —• / / J - ^ 21/J ^* A —• 0