SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS
15
given in [BC2, §5] to show that the equivalent conditions of Theorem 1.11 are not always
satisfied.
Now consider the case in which 21 is a commutative Banach algebra with radical 9t
such that 21/91 = C(Q) and 91 is nilpotent (but Q is an arbitrary compact space). For
many years it was an open question whether or not 21 necessarily has a strong Wedderburn
decomposition. This was proved to be the case by Solovej ([Sol], [So2]) in the important
special case in which Q = I, the closed unit interval in R; Solovej also required that
the quotient norm on 2l/9t be exactly equal to the uniform norm on C(ft). Recently
Albrecht and Ermert have modified Solovej's methods to obtain the following theorem
([AlEr]), which fully resolves the open question.
1.12. THEOREM. Let 21 be a commutative Banach algebra with radical 91 such that
21/91 = C(tt) for a compact space ft and such that 91 is nilpotent. Then 21 has a strong
Wedderburn decomposition.
It is important to note that 21 must be both commutative and a Banach algebra in
the above theorem. For let Q be an infinite compact space. In [Sol], Solovej constructs
a singular, commutative algebraic extension of C(Q) which does not split algebraically,
and so the algebra 21 in Theorem 1.12 must be a Banach algebra. Also, as we shall note
in Theorem 3.11(i), there is a singular, admissible (non-commutative) extension of C(Q)
which does not split algebraically.
In the case where 91 is nilpotent and $& is totally disconnected, but 21/91 C C($&),
there is not always a Wedderburn decomposition of 21, for an example, of a commutative,
unital Banach algebra with the following properties is given in [BC2]: (i) $% is totally
disconnected; (ii) 9t3 = 0; (iii) 21 does not have a Wedderburn decomposition.
After the early work of Feldman and of Bade and Curtis, the next important advance
was due to Kamowitz in 1962 ([Kam]), when cohomological methods derived from [Hoi]
and [Ho2] were introduced into Banach algebra theory; Kamowitz concentrated on commu-
tative Banach algebras. The story was taken up by Johnson in 1968 ([Jo2]); in this paper,
the connection between cohomology theory and different types of Wedderburn decomp-
ositions was made explicit, especially in connection with finite-dimensional extensions.
This connection will be described in our §2, and the results of Kamowitz and Johnson will
be contained in our later theorems.
Also in the 1960s, Helemskii obtained important results on strong decompositions of
Banach algebras; see for example [Hel]. This work is clearly explained in [He6, 1.1],
where detailed references are given; in this exposition, the author works with more general
topological algebras than Banach algebras.
There is a very substantial theory of extensions XX^5 I) °f a ^* -algebra in the case
where 21 and / are also both C* -algebras. We shall not discuss this theory at all: see
[Do] for results up to 1980, and [Bl] and [W-O] for later results.
There is a natural situation in which non-semisimple Banach algebras arise. Let A be
a regular Banach function algebra on its character space $A and let E be a closed subset
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