Quasi-equivalent representations

Two unitary representations (cf. Unitary representation) of a group (or symmetric representations of a symmetric algebra ) in Hilbert spaces and , respectively, satisfying one of the following four equivalent conditions: 1) there exist unitarily-equivalent representations and such that is a multiple of and is a multiple of ; 2) the non-zero subrepresentations of are not disjoint from , and the non-zero subrepresentations of are not disjoint from ; 3) is unitarily equivalent to a subrepresentation of some multiple representation of that has unit central support; or 4) there exists an isomorphism of the von Neumann algebra generated by the set onto the von Neumann algebra generated by the set such that for all . Unitarily-equivalent representations are quasi-equivalent representations; irreducible quasi-equivalent representations (cf. Irreducible representation) are unitarily equivalent. If and are quasi-equivalent representations and is a factor representation, then so is ; a factor representation and a non-zero subrepresentation of it are quasi-equivalent representations; two factor representations are either disjoint or quasi-equivalent. The notion of quasi-equivalent representations leads to that of a quasi-dual object and a quasi-spectrum for locally compact groups and symmetric algebras, respectively.

References

[1]	J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)

Comments

Two representations and (of a group or algebra) with representation spaces and , respectively, are said to be disjoint is there is no non-zero intertwining operator between and . Here, an intertwining operator between and is a continuous linear operator such that for all .