Let H be the Hilbert space completion of with respect to the inner product and let J denote the extension of the involution to a conjugate-linear involution of H. Define a representation λ and an anti-representation ρ of on itself by left and right multiplication:
These actions extend continuously to actions on H. In this case the commutation theorem for Hilbert algebras states that
the von Neumann algebra generated by the operators λ(a), then
These results were proved independently by Godement (1954) and Segal (1953).
The proof relies on the notion of "bounded elements" in the Hilbert space completion H.
An element of x in H is said to be bounded (relative to ) if the map a → xa of into H extends to a bounded operator on H, denoted by λ(x). In this case it is straightforward to prove that:
- Jx is also a bounded element, denoted x*, and λ(x*) = λ(x)*;
- a → ax is given by the bounded operator ρ(x) = Jλ(x*)J on H;
- M ' is generated by the ρ(x)'s with x bounded;
- λ(x) and ρ(y) commute for x, y bounded.
The commutation theorem follows immediately from the last assertion. In particular
- M = λ".
The space of all bounded elements forms a Hilbert algebra containing as a dense *-subalgebra. It is said to be completed or full because any element in H bounded relative to must actually already lie in . The functional τ on M+ defined by
if x =λ(a)*λ(a) and ∞ otherwise, yields a faithful semifinite trace on M with
There is a one-one correspondence between von Neumann algebras on H with faithful semifinite trace and full Hilbert algebras with Hilbert space completion H.
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