By Robert S. Doran, Josef Wichmann (auth.)
ISBN-10: 3540097252
ISBN-13: 9783540097259
ISBN-10: 3540385339
ISBN-13: 9783540385332
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Extra resources for Approximate Identities and Factorization in Banach Modules
Sample text
Max li~ilThen A has pointwise-bounded approximate units u. = ( i , .... 1 , 0 , 0 .... ). 7) A has no bounded approximate identity. Theorem. A commutative Banach algebra bounded approximate units if and only if A A has pointwise- has a bounded approximate identity. Proof. Let A be a commutative Banach algebra with pointwise- bounded approximate units. Define for each n = 1,2,... 56 I. xl = x IDENTITIES IN NO~MED ALGEBRAS for some sequence is a closed subset of Then A with lim xj = x, then there exist sequences n [luijll -< n and ljx- A.
I. such that: be a function, H(m + n) < H(m)H(n), iii) is there a bounded H H + c; and for no equivalent H(n) ii) A {e(n)} of bound there exists a separable Bausch algebra If" ;I '-bound less than or equal to Proof. i. for every ~ > 0, there is an equivalent norm on A in which {e(n)} has bound less than b) bounded H. H > 1 bounded norm interest. H > 1 there exists a separable Banach with a sequential such that for no equivalent here and A used in earlier is 'the Banach algebra e(i)e(j) = e(j) (i > j) = H(m) ' , we mean that the following be the algebra with generators and construction and relations as O stated.
Right, two-sided) if is called a is open. Mr(A) N zr(~) = ~ A A so that the complementary set lim Xna = lim (xn - Xn)a + lim Xn a = 0. i) in zero divisors in is a dense subspace of A z if there is a sequence for all be the completion of seen as follows: sequence A llXnl I = i The set of all right topological Zr(A). An element x A net A is a right identity in with A in A approximate identity in fix - u~xll in {u%} and fix - xu~l I) l lxll = i.
Approximate Identities and Factorization in Banach Modules by Robert S. Doran, Josef Wichmann (auth.)
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