In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint.
A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: Another important class of non-Hilbert C*-algebras includes the algebra
of complex-valued continuous functions on X that vanish at infinity, where X is a locally compact Hausdorff space.
C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables.
This line of research began 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 that are now known as von Neumann algebras.
Around 1943, the work of Israel Gelfand and Mark Naimark 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 unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics.
We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gelfand and Naimark.
A C*-algebra, A, is a Banach algebra over the field of complex numbers, together with a map
For instance, together with the spectral radius formula, it implies that the C*-norm is uniquely determined by the algebraic structure: A bounded linear map, π : A → B, between C*-algebras A and B is called a *-homomorphism if In the case of C*-algebras, any *-homomorphism π between C*-algebras is contractive, i.e. bounded with norm ≤ 1.
[2][3] In his paper Segal defines a C*-algebra as a "uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space".
Some of these properties can be established by using 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.
This partially ordered subspace allows the definition of a positive linear functional on a C*-algebra, which in turn is used to define the states of a C*-algebra, which in turn can be used to construct the spectrum of a C*-algebra using the GNS construction.
In fact, there is a directed family {eλ}λ∈I of self-adjoint elements of A such that 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.
More generally, one can consider finite direct sums of matrix algebras.
In fact, all C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism.
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, 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.
A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics[5] for a finite-dimensional C*-algebra.
The dagger, †, is used in the name because physicists typically use the symbol to denote a Hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions.
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 : 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 Gelfand–Naimark theorem.
To be specific, H is isomorphic to the space of square summable sequences l2; we may assume that H = l2.
of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness) forms a commutative C*-algebra
Such functions exist by the Tietze extension theorem, which applies to locally compact Hausdorff spaces.
The Sherman–Takeda theorem implies that any C*-algebra has a universal enveloping W*-algebra, such that any homomorphism to a W*-algebra factors through it.
This C*-algebra approach is used in the Haag–Kastler axiomatization of local quantum field theory, where every open set of Minkowski spacetime is associated with a C*-algebra.