Heisenberg relation for locally measurable operators
The authors address the question of which operators \(a, b\) on a Hilbert space \(H\) fulfill the Heisenberg relation in the sense that their commutator equals the unit: \([a,b]=1\). It is (well)known that such a relation is not possible if the two operators are bounded. Therefore, it is natural to consider the set of operators affiliated with a von Neumann algebra \(\mathcal{M}\). But since in general this set is not an algebra, the authors are naturally led to consider the smaller set of locally measurable operators \(LS(\mathcal{M})\) because \(LS(\mathcal{M})\) has the maximality property to include each unital \(^*\)-algebra of operators affiliated with \(\mathcal{M}\) and containing \(\mathcal{M}\) (where the algebraic structure is given by the so-called strong addition and strong multiplication). If \(\mathcal{M}\) is the Lebesgue space \(L^\infty(0,\infty)\) (or if \(\mathcal{M}\) is finite or has finite-dimensional centre), then \(LS(\mathcal{M})=S(\mathcal{M})\), where \(S(\mathcal{M})\) denotes the measurable operators affiliated with \(\mathcal{M}\) whereas, if \(\mathcal{M}\) is of type III with infinite-dimensional centre, then \(LS(\mathcal{M})\neq S(\mathcal{M})=\mathcal{M}\). (An operator \(x\) affiliated with \(\mathcal{M}\) is called measurable if \(\chi_{(N,\infty]}(|x|)\) is a finite projection for some \(N\), and it is called locally measurable if the centre of \(\mathcal{M}\) contains a sequence \((z_n)\) increasing to the unit such that \(z_n(H)\) is in \(\text{dom}(x)\) and \(xz_n\in S(\mathcal{M})\).) The first of the two main results recovers and improves, by elementary methods, known results, to wit: For \(a,b\in LS(\mathcal{M})\), \(\mathcal{M}\) a von Neumann algebra, each of the following three conditions implies \([a,b]\neq1\): \(\mathcal{M}\) is infinite or \(\mathcal{M}\) is a finite type I algebra or \(a\) is normal. The second main result is motivated by the known fact that, for \(a,x\in LS(\mathcal{M})\) (\(\mathcal{M}\) a von Neumann algebra), the two equalities \([a,[a,x]]=0\) and \([a,x]=0\) are equivalent. It reads: if \(\mathcal{M}\) is a von Neumann algebra on a separable Hilbert space, if \(x\in \mathcal{M}\) and if \(a_1, \dots, a_d\) are commuting self-adjoint elements of \(LS(\mathcal{M})\) such that \(\sum_{k=1}^d[a_k,[a_k,x]]=0\), then all \(a_k\) commute with \(x\). The authors use devices such as the ideal of generalised compact operators of a von Neumann algebra [\textit{V. Kaftal}, Indiana Univ. Math. J. 26, 447--457 (1977; Zbl 0329.46062)], as well as more recent results on \(S(\mathcal{M})\) and \(LS(\mathcal{M})\).
scientific article; zbMATH DE number 6929150
scientific article