Difference between revisions of "Disjunctive representations"
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''disjoint representations'' | ''disjoint representations'' | ||
| − | Unitary 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 | + | 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 | + | 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==== | ====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> | <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==== | ====Comments==== | ||
| − | |||
====References==== | ====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> | <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> | ||
Revision as of 19:36, 5 June 2020
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, " algebras" , North-Holland (1977) (Translated from French) |
Comments
References
| [a1] | W. Arveson, "An invitation to  -algebras" , Springer (1976) |
How to Cite This Entry:
Disjunctive representations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_representations&oldid=14462
Disjunctive representations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disjunctive_representations&oldid=14462
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article