C*-algebra

C∗-algebras (pronounced "C-star") are subjects of research in, a branch of. A C*-algebra is a together with an involution satisfying the properties of the. A particular case is that of a  A of s on a   with two additional properties:


 * A is a topologically in the  of operators.
 * A is closed under the operation of taking s of operators.

Another important class of non-Hilbert C*-algebra include the algebra of continuous functions $$C_0(X)$$.

C*-algebras were first considered primarily for their use in to  algebras of physical s.  This line of research began with 's  and in a more mathematically developed form with  around 1933. Subsequently, 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 s.

Around 1943, the work of and  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 s of s, 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 s.

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 over the field of s, together with a  $ x \mapsto x^* $  for $ x\in A$      with the following properties:


 * It is an, for every x in A:
 * $$ x^{**} = (x^*)^* = x $$


 * For all x, y in A:
 * $$ (x + y)^* = x^* + y^* $$
 * $$ (x y)^* = y^* x^*$$


 * For every complex number λ in C and every x in A:
 * $$ (\lambda x)^* = \overline{\lambda} x^* .$$


 * For all x in A:
 * $$ \|x^* x \| = \|x\|\|x^*\|.$$

Remark. The first three identities say that A is a. The last identity is called the C* identity and is equivalent to:

$$\|xx^*\| = \|x\|^2,$$

which is sometimes called the B*-identity. For history behind the names C*- and B*-algebras, see the section below.

The C*-identity is a very strong requirement. For instance, together with the, it implies that the C*-norm is uniquely determined by the algebraic structure:


 * $$ \|x\|^2 = \|x^* x\| = \sup\{|\lambda| : x^* x - \lambda \,1 \text{ is not invertible} \}.$$

A, π : A → B, between C*-algebras A and B is called a *-homomorphism if


 * For x and y in A


 * $$ \pi(x y) = \pi(x) \pi(y) \,$$


 * For x in A


 * $$ \pi(x^*) = \pi(x)^* \,$$

In the case of C*-algebras, any *-homomorphism π between C*-algebras is, i.e. bounded with norm ≤ 1. Furthermore, an injective *-homomorphism between C*-algebras is. 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 s that satisfy the condition:


 * $$\lVert x x^* \rVert = \lVert x \rVert ^2$$ for all x in the given B*-algebra. (B*-condition)

This condition automatically implies that the *-involution is isometric, that is, $$\lVert x \rVert = \lVert x^* \rVert $$. Hence, $$\lVert xx^*\rVert = \lVert x \rVert \lVert x^*\rVert$$, 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 $$\lVert x \rVert = \lVert x^* \rVert$$. 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 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 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.

Self-adjoint elements
Self-adjoint elements are those of the form x=x*. The set of elements of a C*-algebra A of the form x*x forms a closed. This cone is identical to the elements of the form xx*. 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 ; the ordering is usually denoted ≥. In this ordering, a self-adjoint element x of A satisfies x ≥ 0 if and only if the of x  is non-negative, if and only if x = s*s for some s. Two self-adjoint elements x and y of A satisfy x ≥ y if x−y ≥ 0.

This partially ordered subspace allows the definition of a on a C*-algebra, which in turn is used to define the  of a C*-algebra, which in turn can be used to construct the  using the.

Quotients and approximate identities
Any C*-algebra A has an. In fact, there is a directed family {eλ}λ∈I of self-adjoint elements of A such that


 * $$ x e_\lambda \rightarrow x $$
 * $$ 0 \leq e_\lambda \leq e_\mu \leq 1\quad \mbox{ whenever } \lambda \leq \mu. $$


 * 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 , i.e. a positive element h such that hAh is dense in A.

Using approximate identities, one can show that the algebraic of a C*-algebra by a closed proper two-sided, with the natural norm, is a C*-algebra.

Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra.

Finite-dimensional C*-algebras
The algebra M(n, C) of n × n over C becomes a C*-algebra if we consider matrices as operators on the Euclidean space, Cn, and  use the  ||&middot;|| on matrices. The involution is given by the. More generally, one can consider finite 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, from which fact one can deduce the following theorem of type:

Theorem. A finite-dimensional C*-algebra, A, is  isomorphic to a finite direct sum
 * $$ A = \bigoplus_{e \in \min A } A e$$

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 {dim(e)}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, this vector is the of the K0 group of A.

A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in for a finite-dimensional C*-algebra. The, †, is used in the name because physicists typically use the symbol to denote a , 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, and especially.

An immediate generalization of finite dimensional C*-algebras are the s.

C*-algebras of operators
The prototypical example of a C*-algebra is the algebra B(H) of bounded (equivalently continuous) s defined on a complex H; here x* denotes the  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.

C*-algebras of compact operators
Let H be a infinite-dimensional Hilbert space. The algebra K(H) of s on H is a 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 {Hi}i∈I such that
 * $$ A \cong \bigoplus_{i \in I } K(H_i),$$

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  for K(H) can be developed. To be specific, H is isomorphic to the space of square summable sequences l2; we may assume that H = l2. For each natural number n let Hn be the subspace of sequences of l2 which vanish for indices k ≤ n and let en be the orthogonal projection onto Hn. The sequence {en}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 of B(H) by K(H) is the.

Commutative C*-algebras
Let X be a Hausdorff space. The space $$C_0(X)$$ of complex-valued continuous functions on X that vanish at infinity (defined in the article on ) 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. 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 which applies to locally compact Hausdorff spaces. Any such sequence of functions $$\{f_K\}$$ is an approximate identity.

The states that every commutative C*-algebra is *-isomorphic to the algebra $$C_0(X)$$, where $$X$$ is the space of  equipped with the. Furthermore, if $$C_0(X)$$ is to $$C_0(Y)$$ as C*-algebras, it follows that $$X$$ and $$Y$$ are. This characterization is one of the motivations for the and  programs.

C*-enveloping algebra
Given a Banach *-algebra A with an, there is a unique (up to C*-isomorphism) C*-algebra E(A) and *-morphism π from A into E(A) which is , 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 G. This is defined as the enveloping C*-algebra of the of G. The C*-algebra of G  provides context for general  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.

Von Neumann algebras
s, known as W* algebras before the 1960s, are a special kind of C*-algebra. They are required to be closed in the, which is weaker than the norm topology.

The 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 is type I.

However, if a C*-algebra has non-type I representations, then by results of 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, 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, where every open set of is associated with a C*-algebra.