C-star-algebra

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C*-algebras are studied in functional analysis. They are a theoretical tool in the theory of unitary representations of locally compact groups, and are also used in the so-called algebraic formulations of quantum mechanics. A C*-algebra A is a Banach algebra over the field of complex numbers, together with a map * : A → A called involution which has the following properties:

(x + y)* = x* + y* for all x, y in A
(λ x)* = λ* x* for every λ in C and every x in A; here, λ* stands for the complex conjugation of λ.
(xy)* = y* x* for all x, y in A
(x*)* = x for all x in A
The C* identity: ||x x*|| = ||x||² for all x in A.

Any C*-algebra is a B*-algebra that is satisifies ||x*|| = ||x|| for all x in A. Not every B*-algebra is a C*-algebra.

A bounded linear map f : A → B between B*-algebras A and B is called a *-homomorphism if

f(xy) = f(x)f(y) for x and y in A
f(x*) = f(x)* for x in A

In the case of C*-algebras, the boundedness condition is superfluous. In fact, any *-homomorphism between C*-algebras is in fact contractive. If f is bijective, then its inverse is also a *-homorphism and f is called a *-isomorphism and A and B are called *-isomorphic.

Table of contents

1 Examples

1.1 Finite dimensional C*-algebras
1.2 C*-algebras of operators
1.3 Commutative C*-algebras
1.4 The C*-algebra of compact operators
1.5 C*-enveloping algebra
1.6 von Neumann algebras

2 C*-algebras and quantum field theory
3 Properties of C*-algebras

Examples

Finite dimensional C*-algebras

The algebra M_n(C) of n-by-n matrices over C becomes a C*-algebra if we consider matrices as operators on the Euclidean space Cⁿ 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.

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 A e 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.

C*-algebras of operators

The prototypical example of a C*-algebra is the algebra L(H) of 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 norm-closed adjoint closed subalgebra of L(H) for a suitable Hilbert space H; this is the content of the Gelfand-Naimark theorem.

Commutative C*-algebras

Let X be a locally compact Hausdorff space. The space of C₀(X) of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*-algebra C₀(X) under pointwise multiplication and addition. The involution is pointwise conjugation. C₀(X) has a multiplicative unit element iff X is compact. As does any C*-algebra, C₀(X) has an approximate identity. In the case of C₀(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-Urysohn 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 an algebra of the form C₀(''X'\').

The C*-algebra of compact operators

Let H be a separable infinite dimensional Hilbert space. K(H) is the algebra of compact operators on H. It is a norm closed subalgebra of L(H). K(H) is also closed under involution; hence it is a C*-algebra. Though K(H) does not have an identity element; an approximate identity for K(H) can be easily displayed. To be specific, let H = l²; for each natural number n let H_n be the subspace of sequences of l² which vanish for indices k ≥ n and let e_n be the orthogonal projection onto H_n. The sequence {e_n}_n is an approximate identity for K(H).

C*-enveloping algebra

Given a B*-algebra A with an approximate identity, there is (up to C*-isomorphism) unique C*-algebra E(A) and *-morphism π from A into E(A) which is universal, that is every other B*-morphism

π': A → B factors uniquely through π. E(A) is called the C*-enveloping algebra of the B*-algebra A.

Of particular importance are the enveloping algebras of the group algebras of locally compact groups. These C*-algebras are used to provide a context for general harmonic analysis.

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 a topology which is weaker than the norm topology. Their study is a specialized area of functional analysis in itself, separate from C*-algebras.

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 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).

See Local quantum physics.

Properties of C*-algebras

C*-algebras have a large number of good technical properties; some of these properties can be established by reduction to commutative C*-algebras. In this case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism theorem or we can use the continuous functional calculus.

Any *-morphism between C*-algebras is contractive.
An injective *-morphism between C*-algebras is isometric.
Any C*-algebra has an approximate identity.
The algebraic quotient of a C*-algebra by a closed proper two-sided ideal is a C*-algebra in a unique way.
The set of self-adjoint elements of a C*-algebra A naturally has the structure of an ordered vector space; the ordering is usually denoted ≤. In this ordering, an element x of A satisfies x ≥ 0 iff x = y* y for some y. We can assume y itself is self-adjoint.