Properties
- Trω(T) is linear in T.
- If T ≥ 0 then Trω(T) ≥ 0
- If S is bounded then Trω(ST) = Trω(TS)
- Trω(T) does not depend on the choice of inner product on H.
- Trω(T) = 0 for all trace class operators T, but there are compact operators for which it is equal to 1.
A trace φ is called normal if φ(sup xα) = sup φ( xα) for every bounded increasing directed family of positive operators. Any normal trace on is equal to the usual trace, so the Dixmier trace is an example of a non-normal trace.
Read more about this topic: Dixmier Trace
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