620

AR Medghalchi

Proof For given e>0, there is a compact set К such that т{хк)> \\m\\ -s Let \1/еСо{Х)^1и10йФй1ф{х)=1опК ТЪ^пф^Хк, 1и{ФШт{хк)^\\т\\-8 So

\\Й^\\т\\ П

As а result of the Lemma 12 we have the following proposition

Proposition 13 The following statements hold

( a ) n-HLc{X)**) = M{X)

( b ) Let F,G€L{Xf'' be such that ||F|| = ||G|| = ||FG|| =1 If FGeLciXf"^, then F, GeLciX)*''

( c ) // F, G are positive functionals of norm one on L(X)* and FGen ^(Co(X)-^), then F, Gen-'(CoiX)^)

Proof (a) Let mG7c~^(Lc(X)**), m^O, and /г be the restriction of 7c(w} to Co(X) Then ||7c(m)|| = ||/i|| and therefore by a simple extension of [1, Theorem 1, p 417], n{m) = fi Since, every meLdX)'^'^ is a linear combination of positive functionals,7r" ^ (Lc(X)**)çM(X)

Conversely , if 1леМ(Х), and m is an extension of fx to L(X)** (by Hahn Banach Theorem), we have II71 (m) 11 = 11 ш|| So ||я(т)|| = ||/г|| Thus 7z(m) = ^ So, the result follows

( b ) Let F = Fi+F2, G = Gi + G2 where F,, G.eLdX)^'', F2, G2e7i-^(Co(X)^) Then, FG = FiGi+(FiG2 + F2Gi + F2G2), and clearly F.G.eLdX)*'' We have also proved that ж~^{Со(Х)^) is an ideal m L(X)**, therefore F1G1+F2G1 + F2G2e7ü-HCo(X)^) Now if FGeLcCX)**, then Fl G2 + F2 Gl+F2G2=0 Thus FG = F,G,, so ||FiGi||=l It follows that ||Fi|| = ||Gi|| = 1, so by theorem 11 F2 = G2 = 0 So F, GeLc(X)**

( c ) Since F and G are positive functionals of norm one, F G is a positive al of norm one So the result follows from (b) □

We have shown m [7] that ЦХ) is an ideal m L(X)** if and only if X IS compact For the general locally compact X we sum up the earlier results m the following mam theorem which gives the structure of Lc(X)**

Tlieorem 14 The following results hold

( a ) ^i(X)çLc(X)**,

( b ) LciXf^ = FLc(X)** e(kerкnLc(X)**),

( c ) FLc(X)** IS isometrically isomorphic with M(X), and f] ЕЬс(Х)*'^=ЦХХ

( d ) ЦХ) is a two sided ideal in LciX)*"^, ^

( e ) L{X) is the topological centre o/Lc(X)**,

( f ) Lc{X)** is a left ideal in L(X)** if and only if X is compact

Proof (a) Let Fg^i(X), ^еЦХ) Then Е^ = цеЦХ)^Ьс{Х)**, so by tion 13 (b), FgLi(X)**

( b ) We know that L(X)** = FL(X)**©ker^, L(J^)** = Lc(X)**e^~'(Co(X)^) so Lc(X)** = ELciX)''* e (ker n n Lc(X)**)

( c ) From Proposition 13 and Theorem 7 it is clear that FLc(X)** = FM(Z), son^bc(X)** = n^M(X)

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Now , we show that f]EM{X) = L{X) If ^eL{X% then E^ = fiE = iaeM(X)

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So fief]EM{X) Conversely, let fiEf]EM{X) Then Ец = 1л for all fieM{X)

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