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

The GNS construction) shows how to interpret every abstract $C^{*}$-algebra as a concrete $C^{*}$-algebra.

$C^{*}$-algebras with a group $G$ that is represented via automorphisms of the algebra are called C-star-systems.

See operator algebras.