# Disjunctive representations

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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.

#### References

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

#### References

 [a1] W. Arveson, "An invitation to $C^*$-algebras" , Springer (1976)
How to Cite This Entry:
Disjunctive representations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_representations&oldid=51351
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article