Disjunctive representations
disjoint representations
Unitary representations of a certain group or, correspondingly, symmetric representations of a certain algebra with an involution which satisfy the following equivalent conditions: 1) the unique bounded linear operator from the representation space of
into the representation space of
is equal to zero; or 2) no non-zero subrepresentations of the representations
and
are equivalent. The concept of disjoint representations is fruitful in the study of factor representations; in particular, a representation
is a factor representation if and only if
cannot be represented as the direct sum of two non-zero disjoint representations. Any two factor representations are either disjoint or else one of them is equivalent to a subrepresentation of the other (and, in the latter case, the representations are quasi-equivalent). The concept of disjoint representations plays an important role in the decomposition of a representation into a direct integral: If
is a representation in a separable Hilbert space
,
is the von Neumann algebra on
generated by the operators of the representation, and
is the centre of
, then
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is the decomposition of the space into the direct integral of Hilbert spaces, which corresponds to the decomposition
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and if also the algebra corresponds to the algebra of diagonalizable operators, then
is a factor representation for almost-all
, and the representations
are pairwise disjoint for almost-all
. There is a simple connection between the disjointness of two representations of a separable locally compact group (or of a separable algebra with an involution) and the mutual singularity of the representatives of canonical classes of measures on the quasi-spectrum of the group (algebra) corresponding to these representations.
References
[1] | J. Dixmier, "![]() |
Comments
References
[a1] | W. Arveson, "An invitation to ![]() |
Disjunctive representations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_representations&oldid=14462