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 {Ì , 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 h» r\*.

EXAMPLE

1.2. Let {Ì , 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: