algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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.
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.
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).
Kehe Zhu, An Introduction to Operator Algebras, CRC Press 1993 (ISBN:9780849378751)
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)
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:
For application to operator K-theory and KK-theory:
See also
Last revised on July 8, 2023 at 16:51:17. See the history of this page for a list of all contributions to it.