nLab
C-star-algebra

Definitions

A $C^{*}$-algebra is a complex Banach algebra over the complex numbers with an involution compatible with complex conjugation (that is a Banach $*$-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} .

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

Concrete $C^{*}$-algebras

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

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

It is immediate that concrete $C^{*}$-algebras correspond precisely to faithful representation?s 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.

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

• Jamie Vicary, Categorical formulation of quantum algebras (arXiv:0805.0432)

Properties

See also operator algebras.

GNS construction

The GNS construction shows that every abstract $C^{*}$-algebra 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 is that of continuous functions on some compact topological space: there is an equivalence of categories $C^{*}{\mathrm{Alg}}_{\mathrm{com}}\simeq$ Top${}_{\mathrm{cpt}}$.

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 $f(C)\subset B$.

Variants

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

Related concepts

• state on an operator algebra

References

A standard textbook reference is chapter 4 in volume 1 of

• Richard Kadison and 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).