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, "![]() |
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
.
Quasi-equivalent representations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-equivalent_representations&oldid=48381