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
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} ,
Sn = linS [ n ] ,
the linear span of £ H
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
J = axA* + + a
A # = {ai&i + + a
: &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 $
} = Q(A).
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