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By B. Renner

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Our s are c o u n t e r e x a m p l e s ferent reason. to the U h l - c o n j e c t u r e It can be shown that the members of ~ for a dif- not only have RNP but also fail the n e i g h b o r l y - t r e e - p r o p e r t y . 6. ON THE N U M B E R OF S E P A R A B L E s Of course, the best known s e p a r a b l e s for K compact metric. ture of these spaces From the i s o m o r p h i c is d e s c r i b e d are the C ( K ) - s p a c e s point of view the struc- in the f o l l o w i n g two results.

That ItG(As)I I iWm G I + s and G - Jmwm for m ~ m a x that + 2 ensure ~ zr s=l _ e for hence F - (JmWm (m0,n) aE $ ~ IF - J n ~ n F1 m ~ m0 1-a IG-JmwmGl~--~--IGI+ ~ G = F - Jn~n 9 IF - 3m~ m F1 GI + -~-. F - Jn~n 9 F) 4~ Thus also ~ IF - J m ~ m rI ( ~ IF - J n ~ n F 1 + a_~ m b 9 and ~ l-a IF - 3 m ~ m F1 9 ~ D ~ m Since this IF - J n ~ n rank 5. THE UHL The C. 14 thaz separable dual Radon-Nikodgm the space. p. were of only then separable Unfortunately, classes p. ~ and dual W have : Let X be an space such Y contains Proof : Let a map : T i : Y ~ X ~ defined and hence the would that are result RNP of if S of X.

24 : If A 6 S ~ for The "regularity" the following PROPOSITION induction gives some of LP-embeddings immediately ~ < ml' then in s u b s p a c e s of A has (~). L p is u s e d to show fact. 25 : If 2 < p < ~ and Lp embeds in L p, t h e n A fails (~). Proof : Take elementary (hn, k) and (~n,k) as in prop. 14. Consider a (~n,k)- function = E so ~ a to w h i c h we associate the f = F e Notice that since LP-function ha . p > 2 Tlfll2 g llfJl g const. p depending on the e q u i v a l e n c e of (h a ) and the Haar-system in L p.

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