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 ${\displaystyle \mathbb {R} }$).

The set of self-adjoint elements of a C*-algebra A naturally has the structure of a partially orderedvector 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 × nmatrices 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.

Una †-álgebra (o, más explícitamente, un álgebra †-cerrada ) es el nombre que se usa ocasionalmente en física ^{[ 5 ]} para un álgebra C* de dimensión finita. La daga , †, se usa en el nombre porque los físicos suelen usar el símbolo para denotar un adjunto hermitiano y a menudo no se preocupan por las sutilezas asociadas con un número infinito de dimensiones. (Los matemáticos suelen usar el asterisco, *, para denotar el adjunto hermitiano). Las †-álgebras tienen un papel destacado en la mecánica cuántica y, especialmente, en la ciencia de la información cuántica .

Una generalización inmediata de las C*-álgebras de dimensión finita son las C*-álgebras de dimensión aproximadamente finita .

El ejemplo prototípico de un álgebra C* es el álgebra B(H) de operadores lineales acotados (equivalentemente continuos) definidos en un espacio de Hilbert complejo H ; aquí x* denota el operador adjunto del operador x  : H → H. De hecho, toda álgebra C*, A , es *-isomorfa a una subálgebra adjunta cerrada en norma de B ( H ) para un espacio de Hilbert adecuado, H ; este es el contenido del teorema de Gelfand-Naimark .

Sea H un espacio de Hilbert separable de dimensión infinita. El álgebra K ( H ) de operadores compactos en H es una subálgebra cerrada en norma de B ( H ). También es cerrada bajo involución; por lo tanto, es un álgebra C*.

Las C*-álgebras concretas de operadores compactos admiten una caracterización similar al teorema de Wedderburn para las C*-álgebras de dimensión finita:

Teorema. Si A es una C*-subálgebra de K ( H ), entonces existen espacios de Hilbert { H _i } _{i ∈ I} tales que

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

donde la suma directa (C*-) consiste en elementos ( T _i ) del producto cartesiano Π K ( H _i ) con || 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) forms 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 groupG. 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.