6

W. G. BADE, H. G. DALES, Z. A. LYKOVA

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

2

G .4, &i,62 € £ ) .

Now suppose that A and B are Banach algebras. Then A® 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

||a||||a||

w

( « U 2 ) .

Let A be a Banach algebra. Then A® 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 = b® 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® = 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.