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). •