Direct integrals of matrices

Abstract

It is shown that each linear operator on a separable Hilbert space which generates a finite type I von Neumann algebra has, up to unitary equivalence, a unique representation as a direct integral of inflations of mutually unitary inequivalent irreducible matrices. This leads to a simplification of the so-called prime (or central) decomposition and the multiplicity theory for such operators. The concept of so-called p-isomorphisms between special classes of such operators is discussed. All results are formulated in more general settings; that is, for tuples of closed densely defined operators affiliated with finite type I von Neumann algebras.

arXiv comment15 pages

References in corpus (1)

• Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces

Cited by in corpus (1)

• Functional calculus in finite type I von Neumann algebras