6 MASAMICHI TAKESAKI DEFINITION 1.4. The von Neumann algebra generated by ðæ(21) is called the left von Neumann algebra of Sl and denoted by Rz(2l). If 21 is a right Hubert algebra instead, then to each ç Å 21, there corresponds a bounded Operator 7^(77) on ip such that (13') ðÃ(ô?)$ = fr?, î Å 21, ô ? Å 21. It then follows that nr is an anti *-representation of 21. The von Neumann algebra generated by 7rr(2l) is called the right von Neumann algebra of 21 and denoted by Rr(2l). We now continue our study of the left Hilbert algebra 21. DEFINITION 1.5. Á vector 7 7 Å ip is said to be right bounded if sup{||7r,(£)7?||: î Å 21, ||£|| 1} = c +~ . Let 99' be the set of all right bounded vectors. Á vector 1 7 Å ip is right bounded if and only if there exists £ Å L(Sp) such that 6* = *,(*)*?, î Å 21. Such an Operator b is unique, so we write it as 71^(77). It is easy to see that ð^(ô?) belongs to Rz(2l)' and ð,.(99') is a left ideal of Rz(2l)'. We then define a product in 99' by the formula (15) 77^2=^(772)7?!, T?J, T?2 Å 99'. We then set (16) a' = »'npK It turns out that 93' is an algebra and 21' is a right Hilbert algebra under the involution (17) T7^ = F77, 77E2I'. THEOREM 1.6. Rz(21)'= Rr(2l'). Now starting from the right Hilbert algebra 21' of (16), we define the left boundedness of a vector î Å ö similarly and 99 denotes the set of all left bounded vectors of $ . Each î Å 99 gives rise to a bounded Operator ðæ(î ) and ðæ(99) is a left ideal of Rz(2l) containing DEFINITION 1.7. Let (18) 21" = 99 n p * . Then 21" is a left Hubert algebra naturally. If 21 = 21", then 21 is said to be füll. Since R/(2l") = R/(2l) and the modular Operators for 21" and for 21 are the same, we may assume that 21 is füll. Up to this point, the arguments were somewhat formal. The next result is the key to the whole Tomita-Takesaki theory. THEOREM 1.8. / / 21 is á füll left Hilbert algebra with mödular Operator Ä, then for any ù Å C, ù & R + , setting

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