8 W. G. BADE, H. G. DALES, Z. A. LYKOVA
by Phillips' Lemma, c$ is not complemented in £°° . (For an elementary proof of this fact,
see [Why].) It is deduced in [Co] that the extension
0 —• JC(H) —• B{H) B{H)/K(H) —+ 0
is not admissible; here H is an infinite-dimensional Hilbert space, and K(H) is the closed
ideal of compact operators on H. More general results are given in [AWhy] and [Joh].
However, a singular extension which is not admissible is not immediately obvious: a specific
example of such an extension of a semisimple Banach algebra has been constructed by
Yakovlev ([Ya]), where indeed an example of a commutative Banach algebra 21 such that
(rad2l)2 = {0} and rad2l is not complemented in 21 is given. We shall describe this
example at the end of this section.
Let $^(21;1) D e a radical extension of a unital Banach algebra A. Then it is standard
that there is an idempotent p £ 21 such that 7r(p) = CA In this case, ^2(p%Lp;pIp)
is also a radical extension of A, and p is the identity of p2lp; if this extension splits
(respectively, splits strongly), then so does the original extension. Thus it will often be
sufficient to assume that 21 is unital.
Let XX^5-0 D e a singular algebraic extension of an algebra A. Then / is not just an
21-bimodule, but also an A -bimodule with respect to the operations
a - x = bx, x a = xb (x J, a G A), (1.2)
where b G 21 is such that 7r(6) = a; these operations are well defined exactly because
I2
= 0. Conversely, let E be a non-zero A -bimodule, and let JZ
=
X X ^ - 0 ^
e a
singular extension of A such that E is isomorphic to I as an A -bimodule. Then the
sequence 53 is a singular algebraic extension of A by E. Similarly, a singular extension
of a Banach algebra A corresponds to a singular extension of A by a Banach A -bimodule.
1.3 DEFINITION. Let A be a Banach algebra, and let E, F, and G be Banach A-
bimodules. Then
J2 ••
0 E -!U F - ^ G 0
is a short exact sequence if J ] is a short exact sequence of .A-bimodules and if, further,
the connecting bimodule homomorphisms U and V are continuous linear maps. The
short exact sequence J2 ^s admissible if there is a continuous linear map Q : G —* F such
that V o Q = iG.
It follows that, for an admissible sequence, there is a continuous linear map P : F —• E
with P o JJ = %E- Thus an admissible short exact sequence is one that splits in the
category of Banach spaces. It is easy to see that a short exact sequence J2 1S admissible
if and only if the closed subspace U(E) is complemented in F, i.e., if and only if there is
a continuous linear projection of F onto U(E).
Note that, in the case where either E or G is finite-dimensional, the short exact
sequence ]T is automatically admissible because linear subspaces of a Banach space which
are either finite-dimensional or of finite codimension are necessarily complemented.
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