mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Russo–Dye theorem is a result in the field of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

. It states that in a unital

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...

, the closure of the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of the

unitary element In mathematics, an element ''x'' of a *-algebra is unitary if it satisfies x^* = x^. In functional analysis, a linear operator ''A'' from a Hilbert space into itself is called unitary if it is invertible and its inverse is equal to its own adjo ...

s is the closed

unit ball Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...

. The theorem was published by B. Russo and H. A. Dye in 1966.

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

is contained in the closed

of the

s. A more precise result is true for the

of all bounded linear operators on a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

: If ''T'' is such an operator and , , ''T'', , < 1 − 2/''n'' for some integer ''n'' > 2, then ''T'' is the mean of ''n''

unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...

Applications

This example is due to Russo & Dye, Corollary 1: If ''U''(''A'') denotes the

s of a

''A'', then the

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...

of a

linear mapping In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

''f'' from ''A'' to a normed linear space ''B'' is :

\sup_ , , f(U), , .

In other words, the norm of an operator can be calculated using only the unitary elements of the algebra.

Notes

{{DEFAULTSORT:Russo-Dye theorem C*-algebras Theorems in functional analysis Unitary operators

Other formulations and generalizations

Applications

Further reading

Notes