C*-algebra

Self-adjoint elements are those of the form ${\displaystyle x=x^{*}}$. The set of elements of a C*-algebra A of the form ${\displaystyle x^{*}x}$ forms a closed convex cone. This cone is identical to the elements of the form ${\displaystyle xx^{*}}$. 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 ${\displaystyle \geq }$. In this ordering, a self-adjoint element ${\displaystyle x\in A}$ satisfies ${\displaystyle x\geq 0}$ if and only if the spectrum of ${\displaystyle x}$ is non-negative, if and only if ${\displaystyle x=s^{*}s}$ for some ${\displaystyle s\in A}$. Two self-adjoint elements ${\displaystyle x}$ and ${\displaystyle y}$ of A satisfy ${\displaystyle x\geq y}$ if ${\displaystyle x-y\geq 0}$.

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.

Any C*-algebra A has an approximate identity. In fact, there is a directed family {e_λ}_λ∈I of self-adjoint elements of A such that

${\displaystyle xe_{\lambda }\rightarrow x}$

${\displaystyle 0\leq e_{\lambda }\leq e_{\mu }\leq 1\quad {\mbox{ whenever }}\lambda \leq \mu .}$

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.

The algebra M(n, C) of n × 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 C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. 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

${\displaystyle A=\bigoplus _{e\in \min A}Ae}$

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. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language of K-theory, this vector is the positive cone of the K₀ group 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. (Mathematicians usually use the asterisk, *, to denote the Hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.

An immediate generalization of finite dimensional C*-algebras are the approximately finite dimensional C*-algebras.

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.

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 {H_i}_i∈I such that

${\displaystyle A\cong \bigoplus _{i\in I}K(H_{i}),}$

where the (C*-)direct sum consists of elements (T_i) of the Cartesian product Π K(H_i) with ||T_i|| → 0.

Though K(H) does not have an identity element, a sequential approximate identity for K(H) can be developed. To be specific, H is isomorphic to the space of square summable sequences l²; we may assume that 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).

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.

Let X be a locally compact Hausdorff space. The space ${\displaystyle C_{0}(X)}$ of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*-algebra ${\displaystyle C_{0}(X)}$ under pointwise multiplication and addition. The involution is pointwise conjugation. ${\displaystyle C_{0}(X)}$ has a multiplicative unit element if and only if ${\displaystyle X}$ is compact. As does any C*-algebra, ${\displaystyle C_{0}(X)}$ has an approximate identity. In the case of ${\displaystyle C_{0}(X)}$ this is immediate: consider the directed set of compact subsets of ${\displaystyle X}$, and for each compact ${\displaystyle K}$ let ${\displaystyle f_{K}}$ be a function of compact support which is identically 1 on ${\displaystyle K}$. Such functions exist by the Tietze extension theorem, which applies to locally compact Hausdorff spaces. Any such sequence of functions ${\displaystyle \{f_{K}\}}$ is an approximate identity.

The Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra ${\displaystyle C_{0}(X)}$, where ${\displaystyle X}$ is the space of characters equipped with the weak* topology. Furthermore, if ${\displaystyle C_{0}(X)}$ is isomorphic to ${\displaystyle C_{0}(Y)}$ as C*-algebras, it follows that ${\displaystyle X}$ and ${\displaystyle Y}$ are homeomorphic. This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs.

Given a Banach *-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) that is universal, that is, every other continuous *-morphism π ' : A → B factors uniquely through π. The algebra E(A) is called the C*-enveloping algebra of the Banach *-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, 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.

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.