COMPLEX ALGEBRAS 
177 
of the Si are all 1 or not all 1, A is said to be of the first 
or second category, respectively. 
This separation of the two cases is nowise necessary 
in the present theory, but is a convenient one since the 
notations in the first case are much simpler than in 
the second case. Although the later treatment of the 
second case applies to both cases, the prior simple dis 
cussion of the first case will greatly clarify that of the 
second case. 
102. Complex algebras A of the first category. We 
have A =5+N, where A is a direct sum of algebras 
(e x ), , . . . , (eh) of order 1, and 
(16) ej = ei, e i e j =o(i^j), 2ei=e, 
e being the modulus of both A and S. Thus 
h 
If eiNej is not zero, its elements are all linear com 
binations of certain of its elements n z ,n 2 , . . . . , which 
are linearly independent. Since n p = eiXej, where x is 
in A, we have 
(17) eiUp = n p , ek.n p =o(k9 £ i) , n p ej—n p , n p e t — o (t^j), 
for k, t=i, . . . . , h. Any element n p ^o which satis 
fies these conditions (17) is said to have the character 
But if eiNej = o, N has no elements of character 
Write 
CjNCj—Qj 1 CjN Cj ,