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 $\mathbb{1}$.

States

See states.

Related concepts

• operator, linear operator

• bounded linear operator / unbounded linear operator

  • compact operator, Calkin algebra

• operator norm

• C*-algebra

  • Gelfand-Naimark theorem

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

  • KK-theory, nonabelian topology?

• von Neumann algebra

References

Textbook:

• Richard V. Kadison, John R. Ringrose, Fundamentals of the theory of operator algebras

  • Vol I Elementary Theory, Graduate Studies in Mathematics 15, AMS 1997 (ISBN:978-0-8218-0819-1, ZMATH)

  • Vol II Advanced Theory, Graduate Studies in Mathematics 16, AMS 1997 (ISBN:978-0-8218-0820-7)

  • Vol III, Special Topics, Elementary Theory – An Exercise Approach, Springer 1991 (doi:10.1007/978-1-4612-3212-4, )

• Gerard Murphy, $C^\ast$-algebras and Operator Theory, Academic Press 1990 (doi:10.1016/C2009-0-22289-6)

• Masamichi Takesaki, Theory of operator algebras I, II and III (pdf I pdf II ZMATH entry).

• Konrad Schmüdgen, Unbounded operator algebras and representation theory, Operator theory, advances and applications, vol. 37. Birkhäuser, Basel (1990) (doi:10.1007/978-3-0348-7469-4)

• Kehe Zhu, An Introduction to Operator Algebras, CRC Press (1993) [ISBN:9780849378751]

• Bruce Blackadar, Operator Algebras – Theory of $C^\ast$-Algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences 122, Springer (2006) [doi:10.1007/3-540-28517-2]

Lecture notes with an eye towards application in quantum physics:

• Klaas Landsman, Lecture notes on operator algebras (2011) (pdf)

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

• Bruce Blackadar, K-Theory for Operator Algebras, Cambridge University Press 1986, second ed. 1999 (pdf)

See also

• Wikipedia, Operator algebras