A complete set of unitary invariants for operators generating finiteW∗-algebras of type I
Carl M. Pearcy · Pacific Journal of Mathematics · 1962
unitary invariants is determined (Theorem 5).In particular, to each such operator A is attached a countable collection of mutually commuting normal operators Ni(A).Then A is unitarily equivalent to B if and only if there is a unitary isomorphism φ between the respective Hubert spaces which satisfies φN i {A)φ~ι = N^B) for all i.The author wishes to express his appreciation to Professor Arlen Brown for his encouragement and patient criticism during the preparation of this paper.2, n x n matrices* We first obtain the result for n x n matrices, or what is the same thing, for operators on an ^-dimensional (complex) Hubert space.The reader is reminded that a ring of operators, or ΫF*-algebra, is a self-adjoint algebra of operators closed in the weak operator topology acting on a Hubert space.ΫF*-algebras are not assumed to contain the identity operator.Throughout this paper W will denote the free multiplicative semigroup generated by the two free variables x and y.Words in this collection are denoted by w(x, y), and the collection of all words w(x, y) with the property that the sum of the exponents appearing in w(x, y) does not exceed n is denoted by W(n).Also, if A is an operator, the notation WJn) denotes the collection of all operators w(A, A*) with w(x, y) e W(ri).LEMMA 2.1.// A is an operator on an n-dimensional Hilbert space, and d is a positive integer such that every operator in W A (d + 1) is a linear combination of operators in W A (d), then W A (d) spans the ""-algebra V generated by A.