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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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dixmier,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331022.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Arveson,   "An invitation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033310/d03331023.png" />-algebras" , Springer  (1976)</TD></TR></table>