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By Tord H. Ganelius

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Therefore A(ap,n) and A"'(ap,n) are topological algebras under the normal topology. This is a generalization of the properties of (A, B) and (c, c"'). We also study some other important examples of topological algebras with such properties. 2. Definitions and Notation We recall some relevant definitions. Through this section we assume that X is a commutative complex topological algebra with unit element. X is called a locally convex algebra if it is also a locally convex space. 1) for all x, y EX.

2. The inductive limit topology is stronger than the normal topology in AO(ap,n). ON o-DUAL ALGEBRAS 35 PROOF. J· -1 = 1 Let Y E 100 (_1_), then sup IYnap,n n ap,n AI < 00 constant Cp such that 1and therefore IYnlap,n 00 for each n. n Illxll p = Ilxllp Ilyllloo' D In [4] it is proved that an echelon space A(ap,n) is nuclear if, and only if, for each p there exist q > p and a sequence u = (un);:::'=o E II such that ap,n = Un aq,n, n = 0,1, ... 3. If A(ap,n) is nuclear, then AO(ap,n), endowed with the normal topology, is such that AO(ap,n) ~ limloo(-I-) ~ limll(-I-).

Now given a pairwise intersecting family of closed balls in X, since X** is a 'PI-space, they intersect in X**. And since X has 'P-IP, they intersect in X too. (a) {::} (e) is also observed in [12, Section 11J. 7. Let A be a family of subsets of X such that F ~ A. Then, the following are equivalent : (a) X is an Ll -predual with A-IP (b) X is an A-C-subspace of every superspace (c) for every A E A, every pairwise intersecting family of closed balls in X with centres in A intersects. PROOF. (a) =?

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