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

is the decomposition of the space into the direct integral of Hilbert spaces, which corresponds to the decomposition

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, " algebras" , North-Holland (1977) (Translated from French)

References

[a1]	W. Arveson, "An invitation to -algebras" , Springer (1976)