C*-algebra

C*-algebras are an important area of research in functional analysis. A C*-algebra can be defined concretely as a complex algebra A of linear operators on a complex Hilbert space with two additional properties:

• A is closed under the operation 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 in an extremely rudimentary form 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 Gel'fand, Mark Naimark and Irving Segal 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.

 Contents

Abstract characterization

We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gel'fand and Naimark.

A C*-algebra A is a Banach algebra over the field of complex numbers, together with a map * : AA called involution. The image of an element x of A under involution is written x*. Involution has the following properties:

• For all x, y in A:
[itex] (x + y)^* = x^* + y^* \quad [itex]
[itex] (x y)^* = y^* x^*. \quad [itex]
• For every λ in C and every x in A:
[itex] (\lambda x)^* = \overline{\lambda} x^*. [itex]
• For all x in A
[itex] (x^*)^* = x. \quad [itex]
• The C* condition holds for all x in A:
[itex] \|x x^* \| = \|x\|^2. [itex]

Any C*-algebra is automatically a B*-algebra, since the C* condition implies that

[itex] \|x \| = \|x^*\| [itex]

for all x in A. However, not every B*-algebra is a C*-algebra.

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

• For x and y in A
[itex] \pi(x y) = \pi(x) \pi(y). \quad [itex]
• For x in A
[itex] \pi(x^*) = \pi(x)^*. \quad [itex]

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

Examples

Finite-dimensional C*-algebras

The algebra Mn(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

[itex] A = \bigoplus_{e \in \min A } A e[itex]

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 Mdim(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 : HH. 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 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 fK 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. {fK}K is an approximate identity.

The Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra C0(X), where X is the space of characters equipped with the weak* topology. Furthermore if C0(X) is isomorphic to C0(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.

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, H is isomorphic to the space of square summable sequences l2, so we may assume that H = l2. For each natural number n let Hn be the subspace of sequences of l2 which vanish for indices kn and let en be the orthogonal projection onto Hn. The sequence {en}n is an approximate identity for K(H).

The quotient of L(H) by K(H) is the Calkin algebra.

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

Of particular importance is the C*-algebra of a locally compact group G. This is defined as the enveloping C*-algebra enveloping 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 non-abelian. In particular, the dual of locally compact group is defined to 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 a topology which is weaker than the norm topology. Their study is a specialized area of functional analysis in itself.

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 : AC with φ(u u*) > 0 for all uA) such that φ(1) = 1. The expected value of the observable x, if the system is in state φ, is then φ(x).

Properties of C*-algebras

C*-algebras have a large number of properties which are technically convenient. These properties can be established by use 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.

• Any *-morphism between C*-algebras has norm ≤ 1.
• 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 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 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 an partially ordered vector space; the ordering is usually denoted ≥. In this ordering, a self-adjoint element x of A satisfies x ≥ 0 iff x is non-negative. Two self-adjoint elements x and y of A satisfy xy if x - y ≥ 0.
• Any C*-algebra has an approximate identity. In fact, there is a directed family {eλ}λ ∈ I of self-adjoint elements of A such that
[itex] x e_\lambda \rightarrow x [itex]
[itex] 0 \leq e_\lambda \leq e_\mu \leq 1\quad \mbox{ whenever } \lambda \leq \mu. [itex]
In case A is separable, A has a sequential approximate identity.

References

• A. Connes, Noncommutative geometry, Academic Press, 1994. This book is widely regarded as a source of new research material, providing much supporting intuition. ISBN 0-121-85860-X
• J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, 1969. 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.
• G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972. Mathematically rigorous reference which provides extensive physics background.

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