Normal operations on quaternionic Hilbert spaces

Abstract

Simple modifications of standard complex methods are used to obtain a spectral theorem, a functional calculus and a multiplicity theory for normal operators on quaternionic Hilbert spaces. It is shown that the algebra of all operators on a quaternionic Hilbert space is a real ${C^\ast }$-algebra in which (a) every normal operator is unitarily equivalent to its adjoint and (b) every operator in the double commutant of a hermitian operator is hermitian. Unitary representations of locally compact abelian groups in quaternionic Hilbert spaces are studied and, finally, the complete structure theory of commutative von Neumann algebras on quaternionic Hilbert spaces is worked out.