proof of topologically irreducible representations are algebraically irreducible for -algebras
As an immediate consequence of Schur’s Lemma for group representations on a Hilbert space we obtain the following result.
We can now prove the result.
Theorem. Let be a algebra. Assume the -representation of on the Hilbert space is topologically irreducible. Then is algebraically irreducible.
By the Lemma it follows that . Hence . By the double commutant theorem (http://planetmath.org/VonNeumannDoubleCommutantTheorem) every operator in (the unit ball in the set of bounded operators ) belongs to the strong operator closure of (the unit ball in ).
To show the algebraical irreducibility of it is enough to find for two given vectors an element such that holds. Indeed, it is enough to consider the case .
Now construct the rank one approximation () with a corresponding , so that .
Approximate further and choose with .
Proceed by induction with . Choose with . Then we have in and which completes the proof. ∎
|Title||proof of topologically irreducible representations are algebraically irreducible for -algebras|
|Date of creation||2013-03-22 19:04:12|
|Last modified on||2013-03-22 19:04:12|
|Last modified by||karstenb (16623)|