algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
(geometry Isbell duality algebra)
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
A -algebra is a Banach algebra over a topological field (often the field of complex numbers) equipped with an anti-involution compatible with complex conjugation if appropriate (that is: a Banach star-algebra) that satisfies the -identity
or equivalently the -identity
A homomorphism of -algebras is a map that preserves all this structure. For this it is sufficient for it to be a star-algebra homomorphism.
Often one sees the definition without the clause (which should be in the definition of Banach -algebra) that the involution is an isometry (so that , which is key for the equivalence of the and identities). This follows easily from the -identity, while it follows from the -identity after some difficulty.
There are different concepts for the tensor product of -algebras, see for example at spatial tensor product.
-algebras equipped with the action of a group by automorphisms of the algebra are called C-star-systems .
Given a complex Hilbert space , a concrete -algebra on is a -subalgebra? of the algebra of bounded operators on that is closed in the norm topology.
A representation of a -algebra on a Hilbert space is a -homomorphism from to the algebra of bounded operators on .
It is immediate that concrete -algebras correspond precisely to faithful representations of abstract -algebras. It is an important theorem that every -algebra has a faithful representation; that is, every abstract -algebra is isomorphic to a concrete -algebra.
The original definition of the term ‘-algebra’ was in fact the concrete notion; the ‘C’ stood for ‘closed’. Furthermore, the original term for the abstract notion was ‘-algebra’ (where the ‘B’ stood for ‘Banach’). However, we now usually interpret ‘-algebra’ abstractly. (Compare ‘-algebra’ and ‘von Neumann algebra’.)
The notion of -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 for instance (Vicary).
-algebras are monadic over sets. More precisely, the forgetful functor that assigns to each algebra the set of points in its unit ball is monadic. See Pelletier & Rosicky (1993).
See also operator algebras.
The self-adjoint elements in a -algebra
form a partially ordered complex vector space by declaring an element to be “larger” than some if the difference is a normal operator
In particular, the positive elements are exactly the normal operators
(It is (only) this partial order on the underlying complex vector space of that determines which linear functions count as states.)
E.g. Murphy (1990) §2.2, Blackadar (2006) §II.3.1
Discussion in the context of algebraic quantum field theory: Bratteli & Robinson (1979) §2.2.2, Fredenhagen (2003) p. 6.
The Gelfand-Naimark theorem says that every C*-algebra is isomorphic to a -algebra of bounded linear operators on a Hilbert space. In other words, every abstract -algebra may be made into a concrete -algebra.
The Gelfand-Naimark-Segal construction (GNS construction) establishes a correspondence between cyclic -representations of -algebras and certain linear functionals (usually called states) on those same -algebras. The correspondence comes about from an explicit construction of the *-representation from one of the linear functionals (states).
Gelfand duality says that every (unital) commutative -algebra over the complex numbers is that of complex-valued continuous functions from some compact Hausdorff topological space: there is an equivalence of categories Top.
Accordingly one may think of the study of non-commutative -algebras as non-commutative topology.
For and two -algebras and a star-algebra homomorphism the set-theoretic image is a -subalgebra of , hence is also the image of in .
This is (KadisonRingrose, theorem 4.1.9).
There is a functor
to the category Poset of posets, which sends each to its poset of commutative subalgebras and sends each morphism to the functor which sends a commutative subalgebra to .
Many -algebras arise as groupoid algebras of Lie groupoids. See at groupoid algebra - References - For smooth geometry
There is homotopy theory of -algebras, being a non-commutative generalization of that of Top. (e.g. Uuye 12). For more see at homotopical structure on C*-algebras.
Any algebra of matrices with coefficients in a -algebra is again a -algebra. In particular is a -algebra for all .
For a -algebra and for a locally compact Hausdorff topological space, the set of continuous functions which vanish at infinity is again a -algebra by extending all operations pointwise. (This algebra is unital precisely if is and if is a compact topological space.)
This algebra is denoted
If then one usually just writes
This are the -algebras to which the Gelfand duality theorem applies and which are the default algebras of observables in classical physics (for a phase space, cf. eg. Landsman (2017), §3).
The subalgebra of compactly supported among all vanishing at infinity-functions (Exp. ) is not in general itself a -algebra, but is a very well-behaved ideal inside , cf. Amini (2004).
A uniformly hyperfinite algebra is in particular a -algebra, by definition.
A von Neumann algebra is in particular a -algebra, by definition.
Monographs:
Jacques Dixmier, Chapter 2 of: -algebras, North Holland (1977) [ch2:pdf, ch13:pdf]
Richard V. Kadison, John R. Ringrose, Fundamentals of the theory of operator algebras, chapter 4 in: Vol I Elementary Theory, Graduate Studies in Mathematics 15, AMS 1997 (ISBN:978-0-8218-0819-1, ZMATH)
Gerard Murphy, -algebras and Operator Theory, Academic Press (1990) [doi:10.1016/C2009-0-22289-6]
Garth Warner, -Algebras, EPrint Collection, University of Washington (2010) [hdl:1773/16302, pdf, pdf]
Ian Putnam, Lecture notes on -algebras (2019) [pdf, pdf]
Bruce Blackadar, Operator Algebras – Theory of -Algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences 122, Springer (2006) [doi:10.1007/3-540-28517-2]
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
See also:
For operator algebra-theory see there and see
Stanisław Woronowicz, Unbounded elements affiliated with -algebras and non-compact quantum groups. Commun. Math. Phys. 136, 399–432 (1991)
Stanisław Woronowicz, K. Napiórkowski, Operator theory in the C*-algebra framework, Reports on Mathematical Physics Volume 31, Issue 3, June 1992, Pages 353–371 (publisher, pdf)
On category-theoretic properties:
A characterizations of injections of commutative sub--algebras – hence of the poset of commutative subalgebras of a -algebra – is in
General properties of the category of -algebras are discussed in
Specifically pullback and pushout of -algebras is discussed in
See also
The homotopy theory of -algebras (a category of fibrant objects-structure on ) is discussed in
For more along such lines see the references at KK-theory and E-theory.
Discussion of -algebras as algebras of observables in quantum physics/quantum probability theory:
Last revised on August 21, 2024 at 02:09:55. See the history of this page for a list of all contributions to it.