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C*-algebra

C*-algebra

C*-algebras (pronounced "C-star") 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:

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 Israel Gelfand, 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.

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:

(x + y)^* = x^* + y^* quad

(x y)^* = y^* x^*. quad

  • For every λ in C and every x in A:

(lambda x)^* = overline{lambda} x^*.

  • For all x in A

(x^*)^* = x. quad

  • The C*–identity holds for all x in A:

|x^* x | = |x|^2.

Note that the C* identity is equivalent to: for all x in A:

|x x^* | = |x|^2.

Thus any C*-algebra is automatically a Banach*-algebra. However, not every Banach*-algebra is a C*-algebra.

The C*-identity is a very strong requirement. For instance, together with the spectral radius formula, it implies the C*-norm is unique.

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

  • For x and y in A

pi(x y) = pi(x) pi(y). quad

  • For x in A

pi(x^*) = pi(x)^*. quad

In the case of C*-algebras, any *-homomorphism π between C*-algebras is non-expansive, i.e. bounded with norm ≤ 1. Furthermore, a *-homomorphism between C*-algebras is isometry. 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.

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. In fact, all finite dimensional C*-algebras are of this form. 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

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 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 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 : HH. 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.

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 if and only if 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 extension 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.

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 {Hi}iI such that A is isomorphic to the following direct sum

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 approximate identity for K(H) can be easily displayed. To be specific, H is isomorphic to the space of square summable sequences l2; we may assume that

H = ell^2. quad

For each natural number n let Hn be the subspace of sequences of l2 which vanish for indices

k geq n

and let

e_n quad

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

C*-enveloping algebra

Given a B*-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 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 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 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. Their study is a specialized area of functional analysis.

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.

  • 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 a partially ordered vector space; the ordering is usually denoted ≥. In this ordering, a self-adjoint element x of A satisfies x ≥ 0 if and only if the spectrum of x is non-negative. Two self-adjoint elements x and y of A satisfy xy if x - y ≥ 0.
  • Any C*-algebra A has an approximate identity. 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 1quad 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 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.

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 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).

See Local quantum physics.

See also

References

  • W. Arveson, An Invitation to C*-Algebra, Springer-Verlag, 1976. ISBN 0387901760. An excellent introduction to the subject, accessible for those with a knowledge of basic functional analysis.
  • A. Connes, Non-commutative geometry, ISBN 0-12-185860-X. This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
  • J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, 1969. ISBN 0720407621. 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. ISBN 0471239003. Mathematically rigorous reference which provides extensive physics background.
  • S. Sakai, C*-algebras and W*-algebras , Springer (1971) ISBN 3-540-63633-1

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