Suppose that A is a non-unital algebra and that E is an A-bimodule. Then E is a unital
A -bimodule for the operations
(ae + a) x = ax + a x , x (ae + a) = ax + x a (a G C, a G A, x G E).
An A -bimodule E over a Banach algebra A is a Banach A-bimodule if it is a Banach
space and if there is a constant C 0 such that
||a-*||C||a||||*|| , ||s-a|| C | | a | | ||s|| (a A, x e E).
By transferring to an equivalent norm on E, we may suppose that C = 1, and we
shall do this throughout. For example, a closed ideal in a Banach algebra A is a Banach
A -bimodule. Again, let E and F be Banach left A -modules over a Banach algebra A.
Then B(E, F) is easily checked to be a Banach A -bimodule with respect to the operations
defined by
(axT)(x) = a-Tx, (T x a)(x) = T(a-x) (x G E) (1.1)
for a G A and T G B(E, F).
Let A and i? be algebras. The (algebraic) tensor product of A and B is denoted by
A 0 5 ; in this algebra,
(ai gbi)(a2 0 62) = aia2®&i&2 (ai,a
G .4, &i,62 £ ) .
Now suppose that A and B are Banach algebras. Then B is a normed algebra with
respect to the projective norm || where {H^,
n n I
the completion of A 0 5 with respect to || H^ is the projective tensor product
(A®B, || H^) of A and B. We also write || H^ for the projective norm on A2: set
Nl* = infi^lKIIII^II : a ^ X A | (aeA2).
[j=l 3=1 J
Clearly we have
( « U 2 ) .
Let A be a Banach algebra. Then A is a Banach A -bimodule with respect to the
module operations defined by the conditions
a (6 0 c) = ab® c, (bS c) a = ca (a,b,c A).
Let i b e a Banach algebra. A net (eA : A G A) in A is a left (respectively, right)
approximate identity if e\a —• a (respectively, aex —• a) for each a G A; a net which
is both a left and a right approximate identity is an approximate identity. Let A be a
Banach algebra with a bounded left approximate identity. Then Cohen's factorization
theorem ([BoDu, 11.11]) asserts that = A.
We now give our formal definition of an algebraic extension; the word 'algebraic' is
inserted to stress the distinction from an 'extension of a Banach algebra', to be defined in
Definition 1.2.
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