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 linear operators on a
complex 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 ...
with two additional properties:
* ''A'' is a topologically
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
in the
norm topology
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
of operators.
* ''A'' is closed under the operation of taking
adjoints of operators.
Another important class of non-Hilbert C*-algebras includes the algebra
of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
Hausdorff space.
C*-algebras were first considered primarily for their use in
quantum mechanics to
model algebras of physical
observables. This line of research began with
Werner Heisenberg's
matrix mechanics and in a more mathematically developed form with
Pascual Jordan around 1933. Subsequently,
John von Neumann attempted to establish a general framework for these algebras, which culminated in a series of papers on
rings of operators. These papers considered a special class of C*-algebras which are now known as
von Neumann algebras.
Around 1943, the work of
Israel Gelfand and
Mark Naimark yielded an abstract characterisation of C*-algebras making no reference to operators on a Hilbert space.
C*-algebras are now an important tool in the theory of
unitary representations of
locally compact groups, and are also used in algebraic formulations of quantum mechanics. Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simple
nuclear C*-algebras.
Abstract characterization
We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gelfand and Naimark.
A C*-algebra, ''A'', is a
Banach algebra over the field of
complex numbers, together with a
map for
with the following properties:
* It is an
involution, for every ''x'' in ''A'':
::
* For all ''x'', ''y'' in ''A'':
::
::
* For every complex number λ in C and every ''x'' in ''A'':
::
* For all ''x'' in ''A'':
::
Remark. The first three identities say that ''A'' is a
*-algebra. The last identity is called the C* identity and is equivalent to:
which is sometimes called the B*-identity. For history behind the names C*- and B*-algebras, see the
history section below.
The C*-identity is a very strong requirement. For instance, together with the
spectral radius formula, it implies that the C*-norm is uniquely determined by the algebraic structure:
::
A
bounded linear map, ''π'' : ''A'' → ''B'', between C*-algebras ''A'' and ''B'' is called a *-homomorphism if
* For ''x'' and ''y'' in ''A''
::
* For ''x'' in ''A''
::
In the case of C*-algebras, any *-homomorphism ''π'' between C*-algebras is
contractive In mathematics, a contraction mapping, or contraction or contractor, on a metric space (''M'', ''d'') is a function ''f'' from ''M'' to itself, with the property that there is some real number 0 \leq k < 1 such that for all ''x'' and ...
, i.e. bounded with norm ≤ 1. Furthermore, an injective *-homomorphism between C*-algebras is
isometric
The term ''isometric'' comes from the Greek for "having equal measurement".
isometric may mean:
* Cubic crystal system, also called isometric crystal system
* Isometre, a rhythmic technique in music.
* "Isometric (Intro)", a song by Madeon from ...
. These are consequences of the C*-identity.
A bijective *-homomorphism ''π'' is called a C*-isomorphism, in which case ''A'' and ''B'' are said to be isomorphic.
Some history: B*-algebras and C*-algebras
The term B*-algebra was introduced by
C. E. Rickart in 1946 to describe
Banach *-algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
s that satisfy the condition:
*
for all ''x'' in the given B*-algebra. (B*-condition)
This condition automatically implies that the *-involution is isometric, that is,
. Hence,
, and therefore, a B*-algebra is also a C*-algebra. Conversely, the C*-condition implies the B*-condition. This is nontrivial, and can be proved without using the condition
. For these reasons, the term B*-algebra is rarely used in current terminology, and has been replaced by the term 'C*-algebra'.
The term C*-algebra was introduced by
I. E. Segal in 1947 to describe norm-closed subalgebras of ''B''(''H''), namely, the space of bounded operators on some Hilbert space ''H''. 'C' stood for 'closed'. In his paper Segal defines a C*-algebra as a "uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space".
Structure of C*-algebras
C*-algebras have a large number of properties that are technically convenient. Some of these properties can be established by using the
continuous functional calculus or by reduction to commutative C*-algebras. In the latter case, we can use the fact that the structure of these is completely determined by the
Gelfand isomorphism In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things:
* a way of representing commutative Banach algebras as algebras of continuous functions;
* the fact that for commutative C*-al ...
.
Self-adjoint elements
Self-adjoint elements are those of the form
. The set of elements of a C*-algebra ''A'' of the form
forms a closed
convex cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every .
...
. This cone is identical to the elements of the form
. Elements of this cone are called ''non-negative'' (or sometimes ''positive'', even though this terminology conflicts with its use for elements of ℝ)
The set of self-adjoint elements of a C*-algebra ''A'' naturally has the structure of a
partially ordered vector space; the ordering is usually denoted
. In this ordering, a self-adjoint element
satisfies
if and only if the
spectrum of
is non-negative, if and only if
for some
. Two self-adjoint elements
and
of ''A'' satisfy
if
.
This partially ordered subspace allows the definition of a
positive linear functional on a C*-algebra, which in turn is used to define the
states of a C*-algebra, which in turn can be used to construct the
spectrum of a C*-algebra using the
GNS construction
GNS may refer to:
Places
* Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia
* Gainesville station (Georgia), an Amtrak station in Georgia, United States
Companies and organizations
* Gesellschaft für Nuklear-Service, a German nuclear-w ...
.
Quotients and approximate identities
Any C*-algebra ''A'' has an
approximate identity. In fact, there is a directed family
λ∈I of self-adjoint elements of ''A'' such that
::
::
: In case ''A'' is separable, ''A'' has a sequential approximate identity. More generally, ''A'' will have a sequential approximate identity if and only if ''A'' contains a
strictly positive element, i.e. a positive element ''h'' such that ''hAh'' is dense in ''A''.
Using approximate identities, one can show that the algebraic
quotient of a C*-algebra by a closed proper two-sided
ideal, with the natural norm, is a C*-algebra.
Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra.
Examples
Finite-dimensional C*-algebras
The algebra M(''n'', C) of ''n'' × ''n''
matrices over C becomes a C*-algebra if we consider matrices as operators on the Euclidean space, C
''n'', and use the
operator norm , , ·, , on matrices. The involution is given by the
conjugate transpose. More generally, one can consider finite
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
s of matrix algebras. In fact, all C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The self-adjoint requirement means finite-dimensional C*-algebras are
semisimple, from which fact one can deduce the following theorem of
Artin–Wedderburn type:
Theorem. A finite-dimensional C*-algebra, ''A'', is canonically isomorphic to a finite direct sum
:
where min ''A'' is the set of minimal nonzero self-adjoint central projections of ''A''.
Each C*-algebra, ''Ae'', is isomorphic (in a noncanonical way) to the full matrix algebra M(dim(''e''), C). The finite family indexed on min ''A'' given by
''e'' is called the ''dimension vector'' of ''A''. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language of
K-theory, this vector is the
positive cone of the ''K''
0 group of ''A''.
A †-algebra (or, more explicitly, a ''†-closed algebra'') is the name occasionally used in
physics[John A. Holbrook, David W. Kribs, and Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction." ''Quantum Information Processing''. Volume 2, Number 5, pp. 381–419. Oct 2003.] for a finite-dimensional C*-algebra. The
dagger, †, is used in the name because physicists typically use the symbol to denote a
Hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the Hermitian adjoint.) †-algebras feature prominently in
quantum mechanics, and especially
quantum information science.
An immediate generalization of finite dimensional C*-algebras are the
approximately finite dimensional C*-algebras.
C*-algebras of operators
The prototypical example of a C*-algebra is the algebra ''B(H)'' of bounded (equivalently continuous)
linear operator
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 ...
s defined on a complex
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 ...
''H''; here ''x*'' denotes the
adjoint operator of the operator ''x'' : ''H'' → ''H''. In fact, every C*-algebra, ''A'', is *-isomorphic to a norm-closed adjoint closed subalgebra of ''B''(''H'') for a suitable Hilbert space, ''H''; this is the content of the
Gelfand–Naimark theorem.
C*-algebras of compact operators
Let ''H'' be a
separable infinite-dimensional Hilbert space. The algebra ''K''(''H'') of
compact operators on ''H'' is a
norm closed subalgebra of ''B''(''H''). It is also closed under involution; hence it is a C*-algebra.
Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*-algebras:
Theorem. If ''A'' is a C*-subalgebra of ''K''(''H''), then there exists Hilbert spaces ''i''∈''I'' such that
:
where the (C*-)direct sum consists of elements (''Ti'') of the Cartesian product Π ''K''(''Hi'') with , , ''Ti'', , → 0.
Though ''K''(''H'') does not have an identity element, a sequential
approximate identity for ''K''(''H'') can be developed. To be specific, ''H'' is isomorphic to the space of square summable sequences ''l''
2; we may assume that ''H'' = ''l''
2. For each natural number ''n'' let ''H
n'' be the subspace of sequences of ''l''
2 which vanish for indices ''k'' ≥ ''n'' and let ''e
n'' be the orthogonal projection onto ''H
n''. The sequence
''n'' is an approximate identity for ''K''(''H'').
''K''(''H'') is a two-sided closed ideal of ''B''(''H''). For separable Hilbert spaces, it is the unique ideal. The
quotient of ''B''(''H'') by ''K''(''H'') is the
Calkin algebra.
Commutative C*-algebras
Let ''X'' be a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
Hausdorff space. The space
of complex-valued continuous functions on ''X'' that ''vanish at infinity'' (defined in the article on
local compactness In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
) form a commutative C*-algebra
under pointwise multiplication and addition. The involution is pointwise conjugation.
has a multiplicative unit element if and only if
is compact. As does any C*-algebra,
has an
approximate identity. In the case of
this is immediate: consider the directed set of compact subsets of
, and for each compact
let
be a function of compact support which is identically 1 on
. Such functions exist by the
Tietze extension theorem, which applies to locally compact Hausdorff spaces. Any such sequence of functions
is an approximate identity.
The
Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra
, where
is the space of
characters equipped with the
weak* topology. Furthermore, if
is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to
as C*-algebras, it follows that
and
are
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
. This characterization is one of the motivations for the
noncommutative topology In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts. The term has its origins in the Gelfand–Naimark theorem, which implies the duality of the category of locally compact Hausd ...
and
noncommutative geometry programs.
C*-enveloping algebra
Given a Banach *-algebra ''A'' with an
approximate identity, there is a unique (up to C*-isomorphism) C*-algebra E(''A'') and *-morphism π from ''A'' into E(''A'') that is
universal, that is, every other continuous *-morphism factors uniquely through π. The algebra E(''A'') is called the C*-enveloping algebra of the Banach *-algebra ''A''.
Of particular importance is the C*-algebra of a
locally compact group ''G''. This is defined as the enveloping C*-algebra of the
group algebra of ''G''. The C*-algebra of ''G'' provides context for general
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
of ''G'' in the case ''G'' is non-abelian. In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*-algebra. See
spectrum of a C*-algebra.
Von Neumann algebras
Von Neumann algebras, known as W* algebras before the 1960s, are a special kind of C*-algebra. They are required to be closed in the
weak operator topology, which is weaker than the norm topology.
The
Sherman–Takeda theorem In mathematics, the Sherman–Takeda theorem states that if ''A'' is 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 o ...
implies that any C*-algebra has a universal enveloping W*-algebra, such that any homomorphism to a W*-algebra factors through it.
Type for C*-algebras
A C*-algebra ''A'' is of type I if and only if for all non-degenerate representations π of ''A'' the von Neumann algebra π(''A'')′′ (that is, the bicommutant of π(''A'')) is a type I von Neumann algebra. In fact it is sufficient to consider only factor representations, i.e. representations π for which π(''A'')′′ is a factor.
A locally compact group is said to be of type I if and only if its
group C*-algebra is type I.
However, if a C*-algebra has non-type I representations, then by results of
James Glimm it also has representations of type II and type III. Thus for C*-algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties.
C*-algebras and quantum field theory
In
quantum mechanics, one typically describes a physical system with a C*-algebra ''A'' with unit element; the self-adjoint elements of ''A'' (elements ''x'' with ''x*'' = ''x'') are thought of as the ''observables'', the measurable quantities, of the system. A ''state'' of the system is defined as a positive functional on ''A'' (a C-linear map φ : ''A'' → C with φ(''u*u'') ≥ 0 for all ''u'' ∈ ''A'') such that φ(1) = 1. The expected value of the observable ''x'', if the system is in state φ, is then φ(''x'').
This C*-algebra approach is used in the Haag-Kastler axiomatization of
local quantum field theory, where every open set of
Minkowski spacetime is associated with a C*-algebra.
See also
*
Banach algebra
*
Banach *-algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
*
*-algebra
*
Hilbert C*-module
*
Operator K-theory
*
Operator system
Given a unital C*-algebra \mathcal , a *-closed subspace ''S'' containing ''1'' is called an operator system. One can associate to each subspace \mathcal \subseteq \mathcal of a unital C*-algebra an operator system via S:= \mathcal+\mathcal ...
, a unital subspace of a C*-algebra that is *-closed.
*
Gelfand–Naimark–Segal construction
*
Jordan operator algebra
Notes
References
* . An excellent introduction to the subject, accessible for those with a knowledge of basic
functional analysis.
* . This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
* . This is a somewhat dated reference, but is still considered as a high-quality technical exposition. It is available in English from North Holland press.
* .
* . Mathematically rigorous reference which provides extensive physics background.
*
* .
*.
{{Authority control
Functional analysis