Banach *-algebra representation
Definition:
A representation of a Banach *-algebra π is a *-homomorphism Ο:πβΆβ¬(H) of π into the *-algebra of bounded operators
on some Hilbert space
H.
The set of all representations of π on a Hilbert space H is denoted rep(π,H).
Special kinds of representations:
-
β’
A subrepresentation of a representation Οβrep(π,H) is a representation Ο0βrep(π,H0) obtained from Ο by restricting to a closed Ο(π)-invariant subspace (http://planetmath.org/InvariantSubspace) 11by a Ο(π)- we a subspace
which is invariant
under every operator Ο(a) with aβπ H0βH.
-
β’
A representation Οβrep(π,H) is said to be nondegenerate if one of the following equivalent
conditions hold:
-
(a)
Ο(x)ΞΎ=0βββxβπβΉΞΎ=0, where ΞΎβH.
-
(b)
H is the closed linear span of the set of vectors Ο(π)H:=
-
(a)
-
β’
A representation is said to be topologically irreducible (or just ) if the only closed -invariant of are the trivial ones, and .
-
β’
A representation is said to be algebrically irreducible if the only -invariant of (not necessarily closed) are the trivial ones, and .
-
β’
Given two representations and , the of and is the representation given by .
More generally, given a family of representations, with , their is the representation , in the direct sum of Hilbert spaces , such that is the direct sum
of the family of bounded operators (http://planetmath.org/DirectSumOfBoundedOperatorsOnHilbertSpaces) .
-
β’
Two representations and of a Banach *-algebra are said to be unitarily equivalent if there is a unitary
such that
-
β’
A representation is said to be if there exists a vector such that the set
is dense (http://planetmath.org/Dense) in . Such a vector is called a cyclic vector
for the representation .
Linked file: http://aux.planetmath.org/files/objects/9843/BanachAlgebraRepresentation.pdf
Title | Banach *-algebra representation |
Canonical name | BanachalgebraRepresentation |
Date of creation | 2013-03-22 17:27:37 |
Last modified on | 2013-03-22 17:27:37 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 23 |
Author | asteroid (17536) |
Entry type | Definition |
Classification | msc 46H15 |
Classification | msc 46K10 |
Defines | subrepresentation |
Defines | cyclic representation |
Defines | cyclic vector |
Defines | nondegenerate representation |
Defines | topologically irreducible |
Defines | algebrically irreducible |
Defines | direct sum of representations |
Defines | unitarily equivalent |