SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS 9

It is clear that an extension $1(21,7") of an algebra A splits algebraically if and only

if there is a subalgebra 03 of 21 with 21 = 03 0 7 (as a semidirect product), where the

symbol 0 denotes the facts that 03 n 7 = {0} and 03 + 7 = 21. An extension $](2l, 7)

of a Banach algebra A splits strongly if and only if there is a closed subalgebra 03 of 21

with 21 = 03 © 7, where the symbol © implies that 21 = 03 © 7 and that both 03 and

7 are closed subspaces of 21.

In the special case where A is a semisimple algebra, an extension J2 (^-5 rad 21) splits

algebraically if and only if 21 has a Wedderbum decomposition, and 5^(21; rad 21) splits

strongly if and only if 21 has a strong Wedderbum decomposition in a standard termin-

ology (e.g., [BDa2]).

We are interested in the following questions.

Question 1 For which Banach algebras A is it true that every extension of A in a

particular class of extensions splits, either algebraically or strongly?

However, we are particularly interested in the following question, which may be inter-

preted as asking when certain maps have 'automatic continuity' properties.

Question 2 For which Banach algebras A is it true that every extension of A in a

particular class of extensions which splits algebraically also splits strongly?

We wish to note immediately that easy examples show that it is not the case that there

is always a positive answer to these questions.

The first examples of commutative Banach algebras without a Wedderbum decomposi-

tion were given by Bade and Curtis in [BC2, §V]. For an easier example, due to Helemskii,

let ;4(D) be the disc algebra, and let

B = {/ 6 A(D) : /'(0) = 0} .

Then B is a Banach function algebra on B . We shall show on page 69 that there is

a commutative, unital Banach algebra 21 with radical 91 such that dim 9^ = 1 and

2l/fH = B, but such that 21 has no Wedderbum decomposition. Similarly, we shall

describe in Theorem 5.13, on page 102, an easy commutative, radical Banach algebra that

has a one-dimensional, commutative extension that does not split algebraically.

Second, we note that there is a semisimple Banach algebra A with an extension

£ = £ ( 2 1 ; rad 21)

with dim rad 21 = 1 such that JZ splits algebraically, but not strongly. The example

originates with Feldman ([Fe]) and is a special case of Theorem 5.1(h).

Before describing this example, we first record a well-known algebraic remark. Let A

be a commutative algebra, and suppose that A = B © 7, where B is a subalgebra of A

and 7 is an ideal of A with 7 C rad A. Then each idempotent p of A belongs to B. For

suppose that p = b 4- r, where b G B and r E 7. Then

b + r = b2 + 2br + r 2 ,