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

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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.

X n) = m a x { K ( ~ ) ..... K(Xn)}O u I .... ,n. (1 - Ul)Xil J K Given and E > 0, there lax i - uixil i < 55 §9. APPROXI~h~TE UNITS (i + K)n-lll(l Hence, for every finite set > 0 all there is an element i = i .... ,n. ~ The next example shows that we cannot in general assert the existence of a bounded approximate identity. 6) Example. ): ~i complex and ~i = 0 for almost all i} with coordinatewise algebraic operations and norm defined by I I(~l,mE .... ) I! = max li~ilThen A has pointwise-bounded approximate units u.

I complex and ~i = 0 for almost all i} with coordinatewise algebraic operations and norm defined by I I(~l,mE .... ) I! = 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,...

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