4

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

general background in Banach algebra theory, see Bonsall and Duncan ([BoDu]) and

Palmer ([Pa2]), for example.

An arbitrarily specified 'algebra' is a linear, associative algebra over the complex field

C.

An ideal in an algebra is always 'two-sided'. The (Jacobson) radical of an algebra A is

denoted by rad A; by definition, the algebra A is semisimple if rad A = {0} and radical

if rad A = A. Let 5 be a subset of A, and let n G N. Then

S^ = {a\ • • • an : a\,..., an G S} ,

and

Sn = linS [ n ] ,

the linear span of £ H

;

i

n

the case where J is an ideal of A, In is also an ideal.

Let A be an algebra. An ideal / in A is nilpotent if In = {0} for some n G N;

clearly, if rad A is finite-dimensional, then rad A is nilpotent. A character on A is an

epimorphism ip : A —+ C The set of characters on A is the character space, denoted by

$A , and we always suppose that $A has the relative weak- * topology from the algebraic

dual space of A; in this topology, pu —• p in $A if and only if ipv(a) — ip(a) for each

a £ A. The kernel of a character p is denoted by M^ , so that Mp is a maximal modular

ideal of codimension one in A.

The identity of a unital algebra A is denoted by e^ , or sometimes by e. The algebra

formed by adjoining an identity to a non-unital algebra A is denoted by A* , so that

A^ = Ce 0 A; in the case where A is a Banach algebra, A*1 is also a Banach algebra.

For an element a in an algebra A, we write

A(e -a) = {b-ba:b e A};

this notation does not imply that A has an identity.

An ideal I in A is algebraically finitely generated if there exist a\,..., an G / such

that

J = axA* + • • • + a

n

A # = {ai&i + • • • + a

n

6

n

: &i,..., bn G A* } .

Let v4 be a unital algebra. The set of invertible elements in A is denoted by Inv A,

and the spectrum of a G A is

a (a) = {C G C : (e - a £ Inv A} ;

in the case where A is not unital, the spectrum of a is

a(a) = {C G C : (e - a £ Inv A* } U {0} .

In each case, an element a is quasi-nilpotent if a(a) C {0}; the set of quasi-nilpotents of

A is denoted by Q,(A). We have rad A C Q(A). In the case where A is a commutative

Banach algebra,

rad A = {a G A : lim ||a n || 1 / n = 0} = C]{M^ : p G $

A

} = Q(A).