Diagonal ring

of a closed symmetric algebra $R$ of bounded linear operators on a Hilbert space $H$

A commutative symmetric Banach algebra $E$ of operators on $H$ such that $(R\cup E)'=E$. Diagonal algebras are employed in decomposing operator algebras into irreducible ones.

Comments[edit]

In the article above, $(R\cup E)'$ denotes the commutant of the minimal closed symmetric algebra containing $R$ and $E$.

In Western terminology, a diagonal ring is called a diagonal algebra. The notion is due to T. Tomita [a1]. "Diagonal ring" only appears in the first edition of [1] and in the translations based on this edition. In the foreword to the second (first revised) American edition (see [a2]), M.A. Naimark noted that "the theory of Tomita is valid only under the additional assumptions of separability type" and that he therefore preferred to give a discussion "which is closer to the initial simpler theory of von Neumann for the separable case" . For a different notion of diagonal algebra see, e.g., [a3].

References[edit]