SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS 5

Let A be an algebra. The algebra with the same structure as A, but with the opposite

multiplication, is denoted by Aop .

Let E and F be linear spaces (respectively, Banach spaces). Then we write £(E,F)

(respectively, B(E, F)) for the linear space of all linear (respectively, all bounded linear)

maps from E into F; we write C(E) and B{E) for £(E, E) and B(E, E), respectively.

The identity map on E is IE . The range and kernel of T G C(E, F) are denoted by

im T and k e r T ,

respectively. The dual space of a Banach space E is denoted by E'.

Recall that a sequence

• * * • An • -A

n

+ i • A

n

+ 2 * ' ' '

of linear spaces Xn and maps Tn G C(Xn,Xn+\) is exact if

imT

n

= ker T

n +

i (n G Z).

Let A be an algebra, and let E be a linear space. Then E is a left A-module (respectively,

a np/&£ A-module) if there is a bilinear map (a, x) •— • a • x (respectively, (a,x) i— • x • a),

A x E —• E , such that a • (b • x) = ab x (respectively, (x • a) • b = x • (ab)) for

a,b e A and x E E. The space E is an A-bimodule if it is both a left A-module and a

right A-module and if

a • (x • b) = (a • x) • 6 (a, 6 G A, x G £ ) .

For example, an ideal I in A is an A-bimodule with respect to the product in A. Let E

and F be A-bimodules. Then we set

A£A(E, F) = {T G £ ( £ , F) : r ( a • x) = a • Tx, T(x • a)=Tx • a (a G A, x G £) } .

In the case where A is a Banach algebra and E and F are Banach A-bimodules, we set

A

BA{E, F) =

A

£A(E, F) n B(E, F).

Let E be a left A-module. Then

AE = {a-x\aeA, x G E} and AE = lin A • E.

The left module i£ is Ze/t annihilator \i A • E = {0}. Similarly, a right A-module E is

ripta annihilator \i E • A = {0}, and an A-bimodule is annihilator if

A • E = E • A = {0} .

An A-bimodule E is symmetric (or commutative) if

a • x = x • a (a E A, x G £ ) .

A symmetric A-bimodule over a commutative algebra A is termed an A-module.

Let A be a unital algebra, with identity e. Then an A-bimodule E is unital if

e- x = x - e = x (x € E).