criterion for a Banach *-algebra representation to be irreducible
Theorem - Let be a Banach *-algebra, an Hilbert space and the identity operator in . A representation (http://planetmath.org/BanachAlgebraRepresentation) is topologically irreducible if and only if , i.e. if and only if the commutant of consists of scalar multiples of the identity operator.
As is selfadjoint, is a von Neumann algebra.
Suppose . Then the dimension of is greater than one.
It is known that von Neumann algebras of dimension greater than one contain non-trivial projections, so there is a projection such that and .
As , commutes with every operator , that is .
Conversely, suppose that is not an irreducible representation. There exists a closed -invariant subspace different from and .
Let be the projection onto that closed invariant subspace.
Invariance can be expressed as: for every . It follows that
for every .
We conclude that commutes with every element of , i.e. .
|Title||criterion for a Banach *-algebra representation to be irreducible|
|Date of creation||2013-03-22 17:27:43|
|Last modified on||2013-03-22 17:27:43|
|Last modified by||asteroid (17536)|