In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. 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 contin ...
, the closure of the
convex hull
In geometry, the convex hull, 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 of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element.
Definition
Let \mathcal be a *-algebra with unit An element a \in \mathcal is called unitary if In o ...
s is the closed
unit ball
Unit may refer to:
General measurement
* Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law
**International System of Units (SI), modern form of the metric system
**English units, histo ...
.
[
]
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
unit ball
Unit may refer to:
General measurement
* Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law
**International System of Units (SI), modern form of the metric system
**English units, histo ...
is contained in the closed
convex hull
In geometry, the convex hull, 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 of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element.
Definition
Let \mathcal be a *-algebra with unit An element a \in \mathcal is called unitary if In o ...
s.
A more precise result is true for the
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 contin ...
of all
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
s on a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
: 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.
Non-trivial examples include rotations, reflections, and the Fourier operator.
Unitary operators generalize unitar ...
s.
[
]
Applications
This example is due to Russo & Dye,
Corollary 1: If ''U''(''A'') denotes the
unitary element In mathematics, an element of a *-algebra is called unitary if it is invertible and its inverse element is the same as its adjoint element.
Definition
Let \mathcal be a *-algebra with unit An element a \in \mathcal is called unitary if In o ...
s of a
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 contin ...
''A'', then the
norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Normativity, phenomenon of designating things as good or bad
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy), a standard in normative e ...
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 Map (mathematics), mapping V \to W between two vec ...
''f'' from ''A'' to a
normed linear space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
''B'' is
:
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:
Notes
{{DEFAULTSORT:Russo-Dye theorem
C*-algebras
Theorems in functional analysis
Unitary operators