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

ISBN-10: 3540097252

ISBN-13: 9783540097259

ISBN-10: 3540385339

ISBN-13: 9783540385332

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Zero divisors. Here we study approximate units and approximate identities in normed algebras which do not consist entirely of topological zero divisors. Let A be a normed algebra. ri__ig~ topological elements in A zero divisor in with Then Zr(A) Hr(A) = A \ Zr(A) Let A 0. {Xn } Since elements in A is closed in A n such that Assume a A. is in of elements in A Then such that Hr(A) N Zr(A). such that A IlXnl I = 1 and so Definition. llx - u%xll of will be denoted by if Let (resp. u This may be Then there exists a IlXnl I = i and there is a sequence and lim Xna = {x } n lim (xn - Xn ) = 0.

Im) , subject to < ~. We shall need another norm on A. For a typical x s A as in (i), we define llxll I Then for llxjl I ! IIxjl the same expression that i. g E 3 E' = im) I . A, but the norms are not equivalent. x a A, and with With E c Z +, we define E' l(i I ..... ,i m) such for some j; and s(E,x) where the summation such that in (i) for a typical S(E,x) the summation x Z [~(i 1 ..... i. a E J = g'' ~''%(i I ..... ,ilkl,i21, 26 Io APPROXIMATE IDENTITIES " . -',inl, " " n) • .. ,i2k 2 ,i31, ....

For the smallest and for the set of all ordinals less than be the set of all finite non-empty subsets of ordered by inclusion. is there a bounded H. i. of If. 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.

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

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