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 ,
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