# Quantum observables

Let H be a Hilbert space and A, B two non-commuting bounded linear operators. Let Com(A,B) be the set of bounded linear operators C which commute both with A and B.

Question 1 : What is known about Com(A,B) ?

And in the quantum mechanical context :

QUestion 2 : What is known about Com(A,B) when H is a complex Hilbert space, and all the operators are Hermitian ?

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Your title and tagging are confusing, since have nothing to do with the question. –  Wadim Zudilin Jul 4 '10 at 13:55
I've re-tagged the problem as 'linear algebra'. (The problem title is appropriate, but vague; quantum observables are hermitian operators on a Hilbert space.) –  Niel de Beaudrap Jul 4 '10 at 17:59

In both cases, Com(A,B) is a weakly closed algebra (i.e., closed in the weak operator topology). In the second case, it is a *-algebra too, so it is a von Neumann algebra.

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