 C*algebra

C*algebras (pronounced "Cstar") are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*algebra is a complex algebra A of linear operators on a complex Hilbert space with two additional properties:
 A is a topologically closed set in the norm topology of operators.
 A is closed under the operation of taking adjoints of operators.
It is generally believed that 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.
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.
Contents
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, * : A → A, called an involution. The image of an element x of A under the involution is written x*. Involution has the following properties:
 For all x, y in A:
 For every λ in C and every x in A:
 For all x in A
 The C*–identity holds for all x in A:
 Note that the C* identity is equivalent to: for all x in A:
This relation 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 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 nonexpansive, i.e. bounded with norm ≤ 1. Furthermore, an injective *homomorphism between C*algebras is isometric. 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 *algebras 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 .^{[1]} 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 normclosed subalgebras of B(H), namely, the space of bounded operators on some Hilbert space H. 'C' stood for 'closed'.^{[2]}^{[3]}
Examples
Finitedimensional C*algebras
The algebra M_{n}(C) of nbyn 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 sums of matrix algebras. In fact, all C*algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The selfadjoint requirement means finitedimensional C*algebras are semisimple, from which fact one can deduce the following theorem of Artin–Wedderburn type:
Theorem. A finitedimensional C*algebra, A, is canonically isomorphic to a finite direct sum
where min A is the set of minimal nonzero selfadjoint 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 {dim(e)}_{e} is called the dimension vector of A. This vector uniquely determines the isomorphism class of a finitedimensional C*algebra. In the language of Ktheory, this vector is the positive cone of the K_{0} group of A.
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 operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x : H → H. In fact, every C*algebra, A, is *isomorphic to a normclosed 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 infinitedimensional 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 {H_{i}}_{i ∈ I} such that A is isomorphic to the following direct sum
where the (C*)direct sum consists of elements (T_{i}) of the Cartesian product Π K(H_{i}) with T_{i} → 0.
Though K(H) does not have an identity element, a sequential approximate identity for K(H) can be easily displayed. To be specific, H is isomorphic to the space of square summable sequences l^{2}; we may assume that
For each natural number n let H_{n} be the subspace of sequences of l^{2} which vanish for indices
and let
be the orthogonal projection onto H_{n}. The sequence {e_{n}}_{n} is an approximate identity for K(H).
K(H) is a twosided 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 Hausdorff space. The space C_{0}(X) of complexvalued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*algebra C_{0}(X) under pointwise multiplication and addition. The involution is pointwise conjugation. C_{0}(X) has a multiplicative unit element if and only if X is compact. As does any C*algebra, C_{0}(X) has an approximate identity. In the case of C_{0}(X) this is immediate: consider the directed set of compact subsets of X, and for each compact K let f_{K} be a function of compact support which is identically 1 on K. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces. {f_{K}}_{K} is an approximate identity.
The Gelfand representation states that every commutative C*algebra is *isomorphic to the algebra C_{0}(X), where X is the space of characters equipped with the weak* topology. Furthermore if C_{0}(X) is isomorphic to C_{0}(Y) as C*algebras, it follows that X and Y are homeomorphic. This characterization is one of the motivations for the noncommutative topology 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) which is universal, that is, every other continuous *morphism π ' : A → B 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 of G in the case G is nonabelian. 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 implies that any C* algebra has a universal enveloping W* algebra, such that any homomorphism to a W* algebra factors through it.
Properties of C*algebras
C*algebras have a large number of properties that are technically convenient. 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.
 The set of elements of a C*algebra A of the form x*x forms a closed convex cone. This cone is identical to the elements of the form x x*. Elements of this cone are called nonnegative (or sometimes positive, even though this terminology conflicts with its use for elements of R.)
 The set of selfadjoint 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 selfadjoint element x of A satisfies x ≥ 0 if and only if the spectrum of x is nonnegative. Two selfadjoint elements x and y of A satisfy x ≥ y if x  y ≥ 0.
 Any C*algebra A has an approximate identity. In fact, there is a directed family {e_{λ}}_{λ ∈ I} of selfadjoint 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 twosided ideal, with the natural norm, is a C*algebra.
 Similarly, a closed twosided ideal of a C*algebra is itself a C*algebra.
Type for C*algebras
A C*algebra A is of type I if and only if for all nondegenerate 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 nontype 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 field theory, one typically describes a physical system with a C*algebra A with unit element; the selfadjoint 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 Clinear 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).
See Local quantum physics.
See also
 algebra
 associative algebra
 * algebra
 Hilbert C*module
 Ktheory
Notes
 ^ Doran & Belfi 1986, pp. 5–6, [1].
 ^ Doran & Belfi 1986, p. 6, [2].
 ^ Segal 1947
References
 Arveson, W. (1976), An Invitation to C*Algebra, SpringerVerlag, ISBN 0387901760. An excellent introduction to the subject, accessible for those with a knowledge of basic functional analysis.
 Connes, Alain, Noncommutative geometry, ISBN 012185860X, http://www.alainconnes.org/docs/book94bigpdf.pdf. This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
 Dixmier, Jacques (1969), Les C*algèbres et leurs représentations, GauthierVillars, ISBN 0720407621. This is a somewhat dated reference, but is still considered as a highquality technical exposition. It is available in English from North Holland press.
 Doran, Robert S.; Belfi, Victor A. (1986), Characterizations of C*algebras: The GelfandNaimark Theorems, CRC Press, ISBN 9780824775698.
 Emch, G. (1972), Algebraic Methods in Statistical Mechanics and Quantum Field Theory, WileyInterscience, ISBN 0471239003. Mathematically rigorous reference which provides extensive physics background.
 A.I. Shtern (2001), "C* algebra", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/c/c020020.htm
 Sakai, S. (1971), C*algebras and W*algebras, Springer, ISBN 3540636331.
 Segal, Irving (1947), "Irreducible representations of operator algebras", Bulletin of the American Mathematical Society 53 (2): 73–88, doi:10.1090/S000299041947087425.
Categories: C*algebras
 Theoretical physics
Wikimedia Foundation. 2010.