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:

      • for all x, y in A
        • stands for the complex conjugation of λ.
    • for all x, y in A
  • = x for all x in A
    • || = ||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
    • for x in A
      • -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

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 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 : 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 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 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 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).

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 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 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.

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.

  • -algebra has an approximate identity.
      • -algebra in a unique way.
        • y for some y. We can assume y itself is self-adjoint.
    • algebra.