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]

[1] |  M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)

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[a1] |  T. Tomita, "Representations of operator algebras" Math. J. Okayama Univ. , 3 (1954) pp. 147–173
[a2] |  M.A. Naimark, "Normed algebras" , Wolters-Noordhoff (1972) (3rd American ed.)
[a3] |  M. Takesaki, "Theory of operator algebras" , 1 , Springer (1979) pp. 259, 273