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

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4) K Theorem. if and only if Proof. K, Let Let A w I]( I - w)xill A by = 1 (i A such that i = 1 ..... n. i, bounded by the constant A K. u, bounded by a constant be any finite set of elements in u, v g A in in A K. i, bounded by the same constant { ~ , . . ,x n} A normed algebra A UNITS and lull < K, ]Ivll < K Ixi(l - v) ll < ~ I - w = (I - v)(l - u). - v)(l - u)xiX I i lJ I - A. ,n. IIi - u I for all i = I ..... n. Ilw - ew]l < g, and Finally we choose IIw - we llxi - exil I J llx i - wxil J I IXi - WXil < ¢.

3) Corollar~. i. i, has bound (n ~ N). the proof of Theorem The Banach algebras identities, . and H ' ( m + n) < H'(m)H'(n). MK n ~ H(n) = nH n This completes left approximate if Now if, in some renorming, then we have shown that that H' to the original norm, H + ~. H(m + n) < H(m)H(n) = nH n , norm on (The s > O, choose a minimal N s Z + subject to and (H + s) N > NHN; we also suppose s <(/2 - I)H. Then (n ~ 2). 1) have bounded but no abelian bounded left approximate identities. Proof. i.. °e(n) is needed.

Ilkl,i21, 26 Io APPROXIMATE IDENTITIES " . -',inl, " " n) • .. ,i2k 2 ,i31, .... ink for i <. j . n, . ] >- I (i <_ j <_ n) and i jp s E. i < p < k.. i, equivalent to If-If, has a bounded MK. i. For each an element f s > 0 such that nc i I fnc e(m) for m < < n. 2) that, e(m) ll Lemma. for every For every finite set ~ > O, there is a positive C of integers greater than integer N such N, lls(C,fn)[lI< ~, for arbitrarily Proof. integer greater large n. If not, then there exists N, there is a finite than set N, and an integer Define, C(N) ~(N) all n ~ ~(N).