nLab
C-star-algebra

Definitions

Abstract $C^{*}$-algebras

Definition

A $C^{*}$-algebra is a Banach algebra $(A,\parallel -\parallel )$ over a topological field $K$ (often the field $K≔ℂ$ of complex numbers) equipped with an involution $(-{)}^{*}$ compatible with complex conjugation if appropriate (that is: a Banach star-algebra) that satisfies the $C^{*}$-identity

$$\parallel A^{*}A\parallel =\parallel A^{*}\parallel \parallel A\parallel$${\|{A^* A}\|} = {\|{A^*}\|} \, {\|{A}\|}

or equivalently the $B^{*}$-identity

$$\parallel A^{*}A\parallel =\parallel A{\parallel }^2.$${\|{A^* A}\|} = {\|{A}\|^2} \,.

A homomorphism of $C^{*}$-algebras is a map that preserves all this structure. For this it is sufficient for it to be a star-algebra homomorphisms.

$C^{*}$-algebras with these homomorphisms form a category C*Alg.

Remark

There are different concepts for the tensor product of $C^{*}$-algebras, see for example at spatial tensor product.

Remark

$C^{*}$-algebras equipped with the action of a group by automorphisms of the algebra are called C-star-systems .

Concrete $C^{*}$-algebras and $C^{*}$-representations

Definition

Given a complex Hilbert space $H$, a concrete $C^{*}$-algebra on $H$ is a $*$-subalgebra of the algebra of bounded operators on $H$ that is closed in the norm topology.

Definition

A representation of a $C^{*}$-algebra $A$ on a Hilbert space $H$ is a $*$-homomorphism from $A$ to the algebra of bounded operators on $H$.

Remark

It is immediate that concrete $C^{*}$-algebras correspond precisely to faithful representations of abstract $C^{*}$-algebras. It is an important theorem that every $C^{*}$-algebra has a faithful representation; that is, every abstract $C^{*}$-algebra is isomorphic to a concrete $C^{*}$-algebra.

Remark

The original definition of the term ‘$C^{*}$-algebra’ was in fact the concrete notion; the ‘C’ stood for ‘closed’. Furthermore, the original term for the abstract notion was ‘$B^{*}$-algebra’. However, we now usually interpret ‘$C^{*}$-algebra’ abstractly. (Compare ‘$W^{*}$-algebra’ and ‘von Neumann algebra’.)

In $†$-compact categories

The notion of $C^{*}$-algebra can be abstracted to the general context of symmetric monoidal †-categories, which serves to illuminate their role in quantum mechanics in terms of †-compact categories.

For a discussion of this in the finite-dimensional case see for instance (Vicary).

Properties

See also operator algebras.

GNS construction

The GNS construction shows that every abstract $C^{*}$-algebra over the complex numbers as a concrete $C^{*}$-algebra: a subalgebra of an algebra of bounded operators on some Hilbert space.

Gelfand duality

Gelfand duality says that every (unital) commutative $C^{*}$-algebra over the complex numbers is that of complex-valued continuous functions from some compact Hausdorff topological space: there is an equivalence of categories $C^{*}{\mathrm{CAlg}}_{\mathrm{com}}\simeq$ Top${}_{\mathrm{cpt}}$.

Accordingly one may think of the study of non-commutative $C^{*}$-algebras as non-commutative topology.

General

Proposition

For $A$ and $B$ two $C^{*}$-algebras and $f:A\to B$ a star-algebra homomorphism the set-theoretic image $f(A)\subset B$ is a $C^{*}$-subalgebra of $B$, hence is also the image of $f$ in $C^{*}\mathrm{Alg}$.

This is (KadisonRingrose, theorem 4.1.9).

Corollary

There is a functor

$$𝒞:C^{*}\mathrm{Alg}\to \mathrm{Poset}$$\mathcal{C} : C^\ast Alg \to Poset

to the category Poset of posets, which sends each $A\in C^{*}\mathrm{Alg}$ to its poset of commutative subalgebras $𝒞(A)$ and sends each morphism $f:A\to B$ to the functor $𝒞(f):𝒞(A)\to 𝒞(B)$ which sends a commutative subalgebra $C\subset A$ to $f(C)\subset B$.

Construction as groupoid convolution algebras

Many $C^{*}$-algebras arise as groupoid algebras of Lie groupoids. See at groupoid algebra - References - For smooth geometry

Homotopy theory

There is homotopy theory of $C^{*}$-algebras, being a non-commutative generalization of that of Top. (e.g. Uuye 12). For more see at homotopical structure on C*-algebras.

Examples

Example

Any algebra $M_n(A)$ of matrices with coefficients in a $C^{*}$-algebra is again a $C^{*}$-algebra. In particular $M_n(ℂ)$ is a $C^{*}$-algebra for all $n\in \mathbb{N}$.

Example

For $A$ a $C^{*}$-algebra and for $X$ a locally compact Hausdorff topological space, the set of continuous functions $X\to A$ which vanish at infinity is again a $C^{*}$-algebra by extending all operations pointwise. (This algebra is unital precisely if $A$ is and if $X$ is a compact topological space.)

This algebra is denoted

$$C_0(X,A)\in C^{*}\mathrm{Alg}.$$C_0(X,A) \in C^\ast Alg \,.

If $A=ℂ$ then one usually just writes

$$C_0(X)≔C_0(X,ℂ).$$C_0(X) \coloneqq C_0(X, \mathbb{C}) \,.

This are the $C^{*}$-algebras to which the Gelfand duality theorem applies.

Example

A uniformly hyperfinite algebra is in particular a $C^{*}$-algebra, by definition.

Example

A von Neumann algebra is in particular a $C^{*}$-algebra, by definition.

Related concepts

• commutative C-star-algebra, Gelfand duality, noncommutative topology

• nuclear C*-algebra

• von Neumann algebra

• Jordan-Lie-Banach algebra

• dense subalgebra, F-star-algebra

• tensor product of C*-algebras

• crossed product C*-algebra

• Cuntz algebra

• Hilbert C*-module, Hilbert C*-bimodule

• C*-category

• C*-coalgebra, Hopf C*-algebra

• homotopical structure on C*-algebras

  • asymptotic C*-homomorphism, l.m.c.-C*-algebra

  • KK-theory, E-theory

• AQFT

  • state on an operator algebra

References

A standard textbook reference is chapter 4 in volume 1 of

• Richard Kadison, John Ringrose, Fundamentals of the theory of operator algebras Academic Press, (1983)

See also the references at operator algebras.

An exposition that explicitly gives Gelfand duality as an equivalence of categories and introduces all the notions of category theory necessary for this statement is in

• Ivo Dell’Ambrogio, Categories of $C^{*}$-algebras (pdf)

A characterizations of injections of commutative sub-$C^{*}$-algebras – hence of the poset of commutative subalgebras of a $C^{*}$-algebra – is in

• Chris Heunen, Characterizations of categories of commutative $C^{*}$-algebras (arXiv:1106.5942)

Properties of the category of $C^{*}$-algebras are discussed in

• Ralf Meyer, Categorical aspects of bivariant K-theory, (arXiv:math/0702145)

The homotopy theory of $C^{*}$-algebras (a category of fibrant objects-structure on $C^{*}\mathrm{Alg}$) is discussed in

• Otgonbayar Uuye, Homotopy theory for $C^{*}$-algebras (arXiv:1011.2926)

For more along such lines see the references at KK-theory and E-theory.