Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Normal operations on quaternionic Hilbert spaces

Author: K. Viswanath
Journal: Trans. Amer. Math. Soc. 162 (1971), 337-350
MSC: Primary 47.30; Secondary 46.00
MathSciNet review: 0284843
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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.

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Keywords: Quaternions, quaternionic Hilbert spaces, spectral theorem, functional calculus, multiplicity theory, row, column, locally compact abelian groups, unitary representations, character group, $ {C^\ast}$-algebra, von Neumann algebra, measure algebra, symplectic image
Article copyright: © Copyright 1971 American Mathematical Society