Disjunctive representations

disjoint representations

Unitary representations $ \pi _ {1} , \pi _ {2} $ 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 $ \pi _ {1} $ into the representation space of $ \pi _ {2} $ is equal to zero; or 2) no non-zero subrepresentations of the representations $ \pi _ {1} $ and $ \pi _ {2} $ are equivalent. The concept of disjoint representations is fruitful in the study of factor representations; in particular, a representation $ \pi $ is a factor representation if and only if $ \pi $ 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 $ \pi $ is a representation in a separable Hilbert space $ H $, $ \mathfrak A $ is the von Neumann algebra on $ H $ generated by the operators of the representation, and $ Z $ is the centre of $ \mathfrak A $, then

$$ H = \int\limits ^ \oplus H ( l) d \mu ( l) $$

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

$$ \pi = \int\limits ^ \oplus \pi ( l) d \mu ( l) , $$

and if also the algebra $ Z $ corresponds to the algebra of diagonalizable operators, then $ \pi ( l) $ is a factor representation for almost-all $ l $, and the representations $ \pi ( l) $ are pairwise disjoint for almost-all $ l $. 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.

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