SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS
19
as a Banach space. Set q\ = q, and, for k Z + \ { 1 } , let qk be the identity map on
0
f c
G . Then
Q : (uk) h- (qk(uk)), 21 - AG ,
is a continuous linear surjection. Define a product in 21 by setting
(m) (VJ) = I uovo, u0vi + ixi^o , I ^ Z *(w») ® Qj(vj) ' k2
It is immediate to check that 21 is a Banach algebra with respect to this product and that
Q : 21 AG is a continuous epimorphism. The kernel of Q is
terQ = {(0,*(t*i),0, . . . ) : t * i £ 7 } ,
and we identify this space with E. Clearly E2 = {0} , and so rad 21 = E and
^ : 0 E —2l A
G
—0
is a singular extension of AG
It is also immediate that the extension splits algebraically. For let T : G —• F be a
linear map such that qoT = %G Then
(tio,ui,ix2, ... ) "-^ (WO,TMI,M
2
,...)» A ? -2l,
is a splitting homomorphism.
Assume towards a contradiction that ]T is admissible. Then there is a continuous
linear projection P : %L E, and this implies that E is complemented as a Banach
subspace of F. By choosing the original short exact sequence a to be such that E is not
complemented in F, we obtain an example for which ^ is not admissible.
To obtain a commutative example with the same properties, we replace AG by BG ,
and make the analogous change to the Banach algebra 21; in this case we require that
G has the approximation property to ensure that BG is semisimple (or use the injective
tensor product).
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