On the structure of self adjoint algebra of finite strict multiplicity (Q1187600)

A strongly closed algebra \({\mathcal A}\) of operators on a complex Hilbert space \(X\) is an algebra of finite strict multiplicity (a.f.s.m.) if it contains the identity operator and if there are finitely many vectors \(x_ 1,\dots,x_ n\) in \(X\) such that every vector of \(X\) can be expressed as \(A_ 1 x_ 1+\dots+ A_ n x_ n\) with \(A_ i\)'s in \({\mathcal A}\). This paper studies the structure of the commutant of an a.f.s.m. Thus, for example, it is shown that if \({\mathcal A}\) is an a.f.s.m., then its commutant \({\mathcal A}'\) can contain at most finitely many mutually orthogonal projections; if, in addition, \({\mathcal A}\) is selfadjoint, then \(X\) can be decomposed as a finite direct sum \(M_ 1\oplus\dots\oplus M_ p\) such that each \(M_ k\) reduces \({\mathcal A}\) and \({\mathcal A}\mid M_ k\) is strongly dense in the algebra of all operators on \(M_ k\), and, in particular, \({\mathcal A}'\) is finite-dimensional.