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
+ i A
+ 2 * ' ' '
of linear spaces Xn and maps Tn G C(Xn,Xn+\) is exact if
= 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
BA{E, F) =
£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 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).
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