(19) we have (20) (20') STRUCTURE OF FACTORS ã(ù ) = ( 2 | ù | - 2 â â ù ) - 1 / 2 , ( Ä - ù ) " 1 » ' C S , ÉÉôô^ Ä - ù)" 1 »?)« 7(ù)||éôÃ(ô?)||, ç «', ( Ä " 1 - ù ) " 1 » C a ' , ÉÉôô,áÄ-- 1 ù)" 1 ÖH ãßù^Éð^)» , $ e Ç. SKETCH OF THE PROOF IN THE CASE OF 31 = Ml0- Ifl this case, it follows that II' = Ì'î 0 · Every left (resp. right) bounded vector is in 31 (resp 31'). We fix ç = ÷'î0, x' Ì' , and î = - ù ) - 1 ô?. Clearly, î e p * = ^(Ä1/ 2 ). We defme two Operators a0 and b0 on Ì º ï = º ' a s foJlows: a0y%=y% b0y%=y^#, / å « ' . It then follows that for each y\ z ' E M ' (a0y%\z%) = (y'Qz%) = (&y'*z%) = ®(z'*y%f) = (z'Vyt*) = (7'lol2'l # ) = ( 7 ' ? 0 I V % ) . so that AQ D b0 and ig D a0. Therefore, a0 and &0 are both preclosed. Let á = a£*, the closure of a0. It is easy to see that the closed Operator á commutes with M', so that it is affiliated with M. Let á = uh = ku be the left and right polar decomposition. Let K(0, °°) be the algebra of all continuous functions on ] 0, »[ with compact support. For each / K(0, oo), we have kf(k) e Ì so that f{k)% 31 and (/(*)£* = (/(*)*«$„)# = (f(k)ku)*H0 = {u*kf(k))i0 = hf(h)u*%0 = f(h)a%=f(h)b0t 0 =7m#-
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