STRUCTURE OF FACTORS 9 SKETCH OF THE PROOF. lf î 1 2 £%Ã\ £(Ä-1/2), then = «Ä + s)iifff 2 ) = (FsmX) + «ttir?r 2 ) = (f1if2i#) + s((i#rf)#if 2 ) = a 1 i(ir 2 # )#) + i(a#ff)*ir 2 ) = (fJA-'/V^CD/A 1 / 2 ^ ) + s C A - ' / V ^ V A 1 / 2 ^ ! ? ^ = ( / ð , ( è * · ^ " 1 / 2 ß é ÉÀ Ä / 2 ^ ) + s ^ / D V A 1 / 2 ^ ^ - 1 / 2 ^ ) . Thus formula (22) holds for fj, f2 31 ç £(Ä-1/2). It suffices to show that every æ G Ñ(Ä1 / 2 ) ¼{ÄÃ÷É) 2 is approximated by a sequence {fn} in Sl ç ß?(Ä _1/2 ) in the following way: f = IimfB, Ä1/ 2 ^ = limA 1 / 2 f„, Ä"1/ 2 ? = lim Ä "1 / 2 ^ . But choosing a sequence {ç„} in Sl' such that (Äé/2 + Ä - é / 2) ? = l i m A-i/iVn, which is possible because Ä"1/ 2 ?!' is dense in ö , we set f n = (i + A)-iT? ll e(i + A r i a ' c a . It is then easy to check that ç } approximates î in the above sense. Q.E.D. Now, we want to express /ð7(î)* / of (22) by an integral of A~ltnr^)Ait. To this end, we prepare the following lemma. LEMMA 1.10. Let Á be á unital Banach algebra and {«(á): á G C} á complex Para- meter group ofinvertible elements in A, i.e. u(a + jS) = u(a)u(0), á, 0 G C. Furthermore, we assume that a G C n u(a) G Á is entire and sup{||w(i)ll: tER} = j|f +«. Then, for any s E R , e~sf2u(- i/2) + es/2u(i/2) is invertible and J e,—™* _~ —«(r)A. °° o^* _L 0 ITT
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