Abstract:
In the articles of M. A. Na\v{i}mark a study has been made of commutative symmetric algebras of bounded operators in the space $\mathsf\Pi_k$. A model was constructed within whose framework all these algebras can be realized, and conditions were found for their unitary equivalence. In the present article a study is made of Na\v{i}ark's model, and some alterations are proposed which lead to a number of important simplifications. We single out the classes of so-called singular and regular algebras; the study of an arbitrary algebra reduces to the study of these. A complete description is given of regular algebras that are closed with respect to the operator norm.
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