On the Identity of Some von Neumann Algebras and Some Consequences

Abstract

Among von Neumann algebras, the Weyl algebra ${\mathcal{W}}$ generated by two unitary groups {U(α)} and {V(β)}, the algebra ${\mathcal{U}}$ generated by a completely non-unitary semigroup of isometries {U ₊(α)} and the Weyl algebra ${\mathcal{W}_{+}^{h}}$ pertaining to a semi-bounded space with homogeneous spectrum of the generator of {V(β)}, all share the property that their representations are completely reducible and the irreducible representations are equivalent. We trace this fact to the identity of these algebras, in the sense that any of them contains a representation of any of the remaining two algebras, which in turn contains the original algebra. We prove this statement by explicit construction. The aforementioned results about the representations of the algebras follow immediately from the proof for any of them. Also, by the above construction we prove for ${\mathcal{W}^{h}_{+}}$ the analog of a theorem by Sinai for ${\mathcal{W}}$ : given {V(β)} with semi-bounded homogeneous spectrum, there exists a completely non-unitary semigroup {U ₊(α)} such that {V(β)} and {U ₊(α)} generate ${\mathcal{W}_{+}^{h}}$.