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+}
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