Contents

Contents

Idea

An operator algebra is any subalgebra of the algebra of continuous linear operators on a topological vector space, with composition as the multiplication. In most cases, the space is a separable Hilbert space, and most attention historically has been paid to algebras of bounded linear operators. The operator algebras themselves are often equipped with their own topologies (e.g. norm topology, weak topology, weak$^*$ topology and so on) and sometimes involution. $C^*$-algebras are (in the complex case) norm-closed complex $*$-subalgebras? of the algebra of the bounded linear operators on a Hilbert space. They can be however characterized structurally as a $*$-representation of a Banach algebra with involution satisfying some compatibility conditions; the latter could be called abstract $C^*$-algebras (or sometimes $B^*$-algebras). Every commutative unital $C^*$-algebra can be realized as an algebra of functions on a Hausdorff compact space, which is obtained as its Gel'fand spectrum.

Special class of $C^*$-algebras are von Neumann or $W^*$-algebras (see wikipedia), which are (in the complex case) weakly closed (closed in the weak topology) $*$-subalgebras of the algebra of bounded linear operators on a Hilbert space. In structural theory, an important role among von Neumann algebras is played by so-called factors, characterized by the property that their center is trivial ($1$-dimensional, consisting of “scalars”).

Operator algebras play major roles in functional analysis, representation theory, noncommutative geometry and quantum field theory.

Topics of interest for the understanding of AQFT

This paragraph will collect some facts of interest for the aspects of AQFT in the nLab. Unless stated otherwise all operator algebras will be assumed to be unital, i.e. having an identiy denoted by $\mathbb{1}$.

See states.

References

Textbook accounts

Lecture notes with an eye towards application in quantum physics:

For application to operator K-theory and KK-theory: