4 MASAMICHI TAKESAKI the von Neumann algebra generated by unitaries commuting with the Operators. The natural questions are the following: (A) What does the algebra Ì look like? (B) How does the algebra Ì act on ö ? These two problems are deeply related and problem (B) was essentially reduced to problem (A) in the 50's. Thus, the following revised question arises: (B') How can we reconstruct ip from M? How can we relate functionals ÷ G Ì ð-* (÷î|£), î G ö , to one another? (Â") Is there any auxiliary mathematical structure attached to , î}? This chapter is devoted mainly to question (B'). Question (B") is left to the lecture of Kosaki. 1. Left Hilbert algebras. DEFINITION 1.1. Á left Hilbert algebra is an involutive algebra 21 over C equipped with an inner product satisfying the following conditions: (1) Every î G 31 gives rise to a bounded linear Operator ç G 31 h+ £7? G 31. (2) (îç\î) = (7?|£#f), where the involution of 31 is denoted by î G 31 h* £# G 31. (3) The involution: £ l·-» î # is preclosed. (4) The subalgebra spanned linearly by îô?, denoted 3l2, is dense in 31. Á right Hilbert algebra will be similarly defined. But in this case, the involution is written as ç r\*. EXAMPLE 1.2. Let , Ö} be a von Neumann algebra which admits a vector î 0 G £) such that [Ìî 0 ] = [M't0] = ö . Here [@] means the closed subspace spanned by © for any subset © of ö . Such a vector î 0 is said to be cyclic and separating. In the set 31 = M£0 w e ^efine (5) and consider the inner product inherited from î) . Then 31 is a left Hilbert algebra and î 0 is the unit of 31. EXAMPLE 1.3. Let G be a locally compact group. We denote by ds the left Haar measure and by SG the modular function on G. Let K(G) be the linear space of all contin- uous functions on G with compact support. We then define an algebraic structure and an inner product in K(G) as follows:
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