Canonical commutation relation, traces and affiliated operators
The relation AB-BA=1 has its roots in quantum mechanics and is related to the so called Heisenberg uncertainty principle. It is well known that this relation can not hold in any normed algebra. For the algebra of matrices it can be shown easily by applying the trace to both sides of the equality. However one can find unbounded operators satisfying this relation. It is natural to ask whether A and B may be chosen to lie in some reasonable algebra. The appropriate notion of such algebra goes back to the work of Murray and von Neumann: this is the so called algebra of operators affiliated with a finite von Neumann algebra. Several years ago it was proven in  that it is impossible to realize A,B as operators lying in the algebra of operators affiliated with a finite von Neumann algebra. One can wonder whether it is possible to prove this using the suitable defined trace on such an algebra. We will construct such trace for the case of type I von Neumann algebra and explain what happens in the type II1 case. The talk is based on joint work with P. Niemiec.
- Z. Liu, On some mathematical aspects of the Heisenberg relation, Sci. China 54 (2011), 2427-2452.
- P. Niemiec, A. Wegert, Algebra of operators affiliated with the finite type I von Neumann algebra, Univ. Iagel. Acta Math. 53 (2016), 39-57.