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# Approximate Identities and Factorization in Banach Modules by Robert S. Doran, Josef Wichmann (auth.)

By Robert S. Doran, Josef Wichmann (auth.)

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Ll~(f2e(N) - e(N))¢NI I Hence e(N) O(f 2) ~ [I,N], for, otherwise, in [~(f2e(N) - e(N))~N](N) flf2 < Choose fl g F, f2 e F be such that Then i. ) Now, it is easy to deduce that consider monomials of identity. 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, .

Clearly If M~ ~ My. A, B are normed algebras, then A®B is an algebra with respect to the multiplication defined by ( x l ® Y l ) ( x 2 ® y 2) = XlX2®yly 2. If • fly The greatest cross-norm is always an algebra norm on for most algebras the least cross-norm I['I[% A®B; but is not submultiplicative. Now let us investigate the existence of approximate identities in tensor products. 1) Proposition. Let admissible algebra norm on {e~}~gM and A, B A®B. s {e~f~}(~,v)sMxN is a A @~ B. n Proof. For any t = Z xi®Y i i=l n in A®B, n t - ( e ®fv)t = i=iZ x i ~ Y i - i~ I e~xi ® f~Yi n n Z (xi - e~xi)® Yi + Z x i ® ( Y i - f~Yi ) i=l i=l n Z i=l and so (xi - e~xi)@(Yi - fvYi )' an 48 I.

L l'-bound less than or equal to Proof. e(%n)II = H(n). ~2" e(~)e(p) = e(~) (~ c p) A typical element of A may be written X = ~({~i ..... ,~,n} with no (16) Xj c ~. (i # j), 3 35 §5. NONSEQUENTIAL and the 6({11 .... ,An} ) z l{({x I ..... In})l H(n) < ~. ,Xn} occurs with in x Assume that A ~ in x. has an algebra norm for belongs MK. i. an element f X. 1 We shall A. i. of If. If -bOund at most to Notice that only can occur in any one element of IIxI! < M-llxll' K, and so of if ~({~i .... 'Xn}) # 0.