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E(i n) with • "2 .... ,ij_ ° I J N il,i amongst the monomials of attainable in 7. f2fl . ~ Quotients. i. i, of the quotient algebra norm. J a closed two-sided ideal in {e%}, then A/I {e% + I} A. is a provided with the quotient 43 §7. 1) Proposition. , Proof. {f6 + l}6sA A. i. i, of then, without loss of generality, in A; for, if Y6 E I l Jf6 + III < H such that Denote by A K and is bounded, can be assumed to be bounded the directed set of all pairs A and n = 1,2, . . f~lxi - xi + I I I H, then there is a Yi ~ I f6x y% ~ F ~ = (F,n), where Then for each ~ = (F,n) F is in such that i < 2(I+K)--------~ (i = i .....

Form a bounded multiplicative semigroup in there exists an equivalent algebra norm A. ), which implies that lim sup I IfnlI' = i. This proves the theorem. 4) Theorem. i, of bound i. 5)). 5) Theorem. i. i. {f } n of bound I. Further, if A A has a sequential is complete, then {f } n may be chosen abelian. 4. Banach algebras with s e q u e n t i a l a p p r o x i m a t e identities. 5) to algebras which have only 24 I. APPROXIMATE bounded left approximate IDENTITIES identities. 1) algebra Theorem.

3 shows that ~'i shows that a i ~ @(fj) f e) = ¢. Such a choice n . Then ~. E l ~i ~ O(fj) (j > i ) . i (j < i), and 37 §5. NONSEQUENTIAL APPROXIMATE IDENTITIES Using a notation slightly different from that employed previously, we define, for x E A as in (16) and ordinals s(B I ..... Bn,X) = E' where the summation BI""'Bn < m2' Z' ~({%11 ..... %nk })e(%ll)'''e(%nk )" n n is over those sets which may be labelled {%11,%12 .... > i (I. and fi = ui + vi' where x s A. Xij (i . < i < n, i < j < ki). u n containing one or more v's.