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Difference between revisions of "Diagonal ring"

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''of a closed symmetric algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315301.png" /> of bounded linear operators on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315302.png" />''
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''of a closed symmetric algebra $R$ of bounded linear operators on a Hilbert space $H$''
A commutative symmetric Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315303.png" /> of operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315304.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315305.png" />. Diagonal algebras are employed in decomposing operator algebras into irreducible ones.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
 
 
 
  
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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====
 
====Comments====
In the article above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315306.png" /> denotes the commutant of the minimal closed [[Symmetric algebra|symmetric algebra]] containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315308.png" />.
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In the article above, $(R\cup E)'$ denotes the commutant of the minimal closed [[Symmetric algebra|symmetric algebra]] containing $R$ and $E$.
  
In Western terminology, a diagonal ring is called a diagonal algebra. The notion is due to T. Tomita [[#References|[a1]]].  "Diagonal ring"  only appears in the first edition of [[#References|[1]]] and in the translations based on this edition. In the foreword to the second (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031530/d0315309.png" /> first revised) American edition (see [[#References|[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., [[#References|[a3]]].
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In Western terminology, a diagonal ring is called a diagonal algebra. The notion is due to T. Tomita [[#References|[a1]]].  "Diagonal ring"  only appears in the first edition of [[#References|[1]]] and in the translations based on this edition. In the foreword to the second (first revised) American edition (see [[#References|[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., [[#References|[a3]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Tomita,  "Representations of operator algebras"  ''Math. J. Okayama Univ.'' , '''3'''  (1954)  pp. 147–173</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.A. Naimark,  "Normed algebras" , Wolters-Noordhoff  (1972)  (3rd American ed.)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Takesaki,  "Theory of operator algebras" , '''1''' , Springer  (1979)  pp. 259, 273</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Tomita,  "Representations of operator algebras"  ''Math. J. Okayama Univ.'' , '''3'''  (1954)  pp. 147–173</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.A. Naimark,  "Normed algebras" , Wolters-Noordhoff  (1972)  (3rd American ed.)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Takesaki,  "Theory of operator algebras" , '''1''' , Springer  (1979)  pp. 259, 273</TD></TR></table>

Latest revision as of 11:41, 26 March 2023

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

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

[1] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[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
How to Cite This Entry:
Diagonal ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_ring&oldid=17612
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article