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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