Quadratic unconstrained binary optimization: Difference between revisions

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In [[mathematics]], the '''Russo–Dye theorem''' is a result in the field of [[functional analysis]]. It states that in a [[unital algebra|unital]] [[C*-algebra]], the closure of the [[convex hull]] of the [[unitary element]]s is the closed [[unit ball]].<ref name="Doran Belfi">
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{{cite book | last = Doran | first = Robert S. |coauthors = Victor A. Belfi | title = Characterizations of C*-Algebras: The Gelfand–Naimark Theorems | publisher = Marcel Dekker | location = New York | year = 1986 | isbn = 0-8247-7569-4 }}
</ref>{{Rp|44}}
The theorem was published by B. Russo and H. A. Dye in 1966.<ref name="Russo Dye">
{{cite journal |last=Russo |first=B. |coauthors=H. A. Dye |year=1966|title=A Note on Unitary Operators in C*-Algebras |journal=Duke Mathematical Journal |volume=33 |pages=413–416 |doi=10.1215/S0012-7094-66-03346-1 |issue=2 }}
</ref>
 
==Other formulations and generalizations==
 
Results similar to the Russo–Dye theorem hold in more general contexts. For example, in a unital *-Banach algebra, the closed [[unit ball]] is contained in the closed [[convex hull]] of the [[unitary element]]s.<ref name="Doran Belfi" />{{Rp|73}}
 
A more precise result is true for the [[C*-algebra]] of all [[bounded linear operator]]s on a [[Hilbert space]]: If ''T'' is such an operator and ||''T''|| < 1&nbsp;&minus;&nbsp;2/''n'' for some integer ''n'' > 2, then ''T'' is the mean of ''n'' [[unitary operator]]s.<ref name="Analysis Now">
{{cite book | last = Pedersen | first = Gert K. | title = Analysis Now | publisher = Springer-Verlag | location = Berlin | year = 1989 | isbn = 0-387-96788-5 }}
</ref>{{Rp|98}}
 
==Applications==
 
This example is due to Russo & Dye,<ref name="Russo Dye" /> Corollary 1: If ''U''(''A'') denotes the [[unitary element]]s of a [[C*-algebra]] ''A'', then the [[Operator norm|norm]] of a [[linear mapping]] ''f'' from ''A'' to a [[normed linear space]] ''B'' is
:<math>\sup_{U \in U(A)} ||f(U)||.</math>
In other words, the norm of an operator can be calculated using only the unitary elements of the algebra.
 
==Further reading==
*  An especially simple proof of the theorem is given in: {{cite journal |last=Gardner |first=L. T. |year=1984 |title=An elementary proof of the Russo–Dye theorem |jstor=2044692 |journal=Proceedings of the American Mathematical Society |volume=90 |issue=1|pages=171 |doi=10.2307/2044692 }}
 
==Notes==
<references />
 
{{DEFAULTSORT:Russo-Dye theorem}}
[[Category:C*-algebras]]
[[Category:Theorems in functional analysis]]
[[Category:Unitary operators]]

Latest revision as of 20:13, 21 September 2014

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