Operator algebras and symmetric spaces (Q1322684)

This short and nice expository paper deals with the relation between the geometry of a globally symmetric Riemannian space \(M\) and the spectral theory of certain operators on the Hilbert space \(L_ 2 (M)\). Consider the von Neumann algebra \(I(M)\) generated by the unitary operators in \(L_ 2(M)\) which arise from isometries of \(M\). The author announces an interesting theorem, providing a description of the normal operators affiliated with the commutant \(I(M)'\). As a corollary, he obtains that all normal operators generated by invariant differential expressions are affiliated with \(I(M)\). At the same time, every invariant differential expression is essentially normal.