STRUCTURE OF FACTORS 5 (ttnXs) = LwWt-'sWt, (6) It then follows that K(G) is a left Hilbert algebra. If we define the involution \ in K(G) as (7) t\s) = îÏ" 1 ), then K(G) is a right Hilbert algebra. We now fix a left Hilbert algebra 21 and denote its completion by ip. The involution î — £# is a conjugate linear preclosed densely defmed Operator. Let S be the closure of this Operator. Since the involution is the inverse of itself, S is also the inverse of S, i.e., S = iS-1. Let V# be the domain of S. Next, we denote the adjoint Operator 5* of S by F and its domain by V . The conjugate linearity of S yields that of F and (8) Pili?) = (FTJIS), î G p # , ô ? G pK With Ä = FS, we obtain the polar decomposition of S: (9) S = JAlf2. The identity S = 5 _ 1 implies that (10) JAJ=A~\ (11) S = A~1I2J, F = JA~ll2 = Ä1 / 2 / , (12) / 2 = 1. We note that / is a conjugate linear isometry of ö onto î ) itself. We call / the modular conjugation and Ä the modular Operator. By condition (1), to each î G 21 there corresponds a bounded Operator ðæ(î ) ï ç ip such that (13) 7rf(t)T? = fT?, î G 21, ô G 21. Condition (2) means then (14) ir,ß # ) = itfö*. * e « . Thus, ð7 is a *-representation of 21 on § . Condition (4) means precisely that ð7(21) is non- degenerate on ßñ.

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