symmetric monoidal (∞,1)-category of spectra
A -algebra is a Banach algebra over a topological field (often the field of complex numbers) equipped with an involution compatible with complex conjugation if appropriate (that is: a Banach star-algebra) that satisfies the -identity
or equivalently the -identity
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.
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 J Wick Pelletier & J Rosicky, On the equational theory of -algebras, Algebra Universalis 30:275-284, 1993.
See also operator algebras.
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.
This is (KadisonRingrose, theorem 4.1.9).
There is a functor
Many -algebras arise as groupoid algebras of Lie groupoids. See at groupoid algebra - References - For smooth geometry
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.
A uniformly hyperfinite algebra is in particular a -algebra, by definition.
A von Neumann algebra is in particular a -algebra, by definition.
A standard textbook reference is chapter 4 in volume 1 of
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)
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