SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS 17

Indeed, let E be a Banach space, and, for i G N, let (g) E denote the projective

— 0 ^ 2

tensor product of i copies of E; set (g) E = C. The projective norm in (g) 1? is now

denoted by || • ||.. The natural identification of ( 0 * £ ) 8 ((g)J £ ) with (g)*+j E extends

to a continuous bilinear map

®:

( 0

E

)

x (S)'E)

-^0*

'E

such that \\u 0 v||

i +

• = \\u\li ||i|| • (*x G (g) £ , ^ G (g) £ 1 . The projective tensor algebra

of £ is

0 ^ = | ( i i

J

- ) : t i

i

€ 0 ^ (jG

with the product

(Wi) ® (vj) = { 5 Z Ui 0 Vj : fe € Z + J .

The projective tensor algebra is a Frechet algebra with respect to the sequence (pk : k G N)

of seminorms, where

k

Pk(*) = J2 WujWj (u = M € ®^)

We consider the subalgebra AE of (g)E defined by

AE = U = (UJ) G 0 £ : ||tx|| = JT ll^ll^. oc I

It is easy to check that AE is indeed a unital Banach algebra. The algebra AE contains

the tensor algebra 0 E a s a dense subalgebra.

We claim that the only quasi-nilpotent element in the Banach algebra AE is 0. For

suppose that u = (UJ) G AE \ {0}, and take

k = min{j G Z + : Uj ^ 0} .

For each n G N,

\\n®n\\ ii7/®nii — n?/jr

\\U I I - 11% llnfc ~~ WUk\\k '

and so ||ti 0n || ||ufc||fc 0. By the spectral radius formula, u is not quasi-nilpotent,

establishing the claim. In particular, AE is semisimple.

The algebra AE is commutative only in the case where dim E = 1. The symmetric

algebra \J E over a linear space E is described in [Gre, Chapter 9]; in fact

\lE = ®{\JE:jeZ+}