algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
(geometry $\leftarrow$ Isbell duality $\to$ algebra)
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
A $C^*$-algebra is a Banach algebra $(A, {\|-\|})$ over a topological field $K$ (often the field $K \coloneqq \mathbb{C}$ of complex numbers) equipped with an anti-involution $(-)^\ast$ compatible with complex conjugation if appropriate (that is: a Banach star-algebra) that satisfies the $C^*$-identity
or equivalently the $B^*$-identity
A homomorphism of $C^\ast$-algebras is a map that preserves all this structure. For this it is sufficient for it to be a star-algebra homomorphism.
$C^\ast$-algebras with these homomorphisms form a category C*Alg.
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 ${\|A^*\|} = {\|A\|}$, which is key for the equivalence of the $B^*$ and $C^*$ identities). This follows easily from the $B^*$-identity, while it follows from the $C^*$-identity after some difficulty.
There are different concepts for the tensor product of $C^*$-algebras, see for example at spatial tensor product.
$C^*$-algebras equipped with the action of a group by automorphisms of the algebra are called C-star-systems .
Given a complex Hilbert space $H$, a concrete $C^*$-algebra on $H$ is a $*$-subalgebra? of the algebra of bounded operators on $H$ that is closed in the norm topology.
A representation of a $C^*$-algebra $A$ on a Hilbert space $H$ is a $*$-homomorphism from $A$ to the algebra of bounded operators on $H$.
It is immediate that concrete $C^*$-algebras correspond precisely to faithful representations of abstract $C^*$-algebras. It is an important theorem that every $C^*$-algebra has a faithful representation; that is, every abstract $C^*$-algebra is isomorphic to a concrete $C^*$-algebra.
The original definition of the term ‘$C^*$-algebra’ was in fact the concrete notion; the ‘C’ stood for ‘closed’. Furthermore, the original term for the abstract notion was ‘$B^*$-algebra’ (where the ‘B’ stood for ‘Banach’). However, we now usually interpret ‘$C^*$-algebra’ abstractly. (Compare ‘$W^*$-algebra’ and ‘von Neumann algebra’.)
The notion of $C^*$-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).
$C^*$-algebras are monadic over sets. More precisely, the forgetful functor $\mathbf{C^*Alg}\to\mathbf{Set}$ 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 $C^\ast$-algebra $\mathcal{A}$
form a partially ordered complex vector space by declaring an element $A$ to be “larger” than some $B$ 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 $\mathcal{A}$ that determines which linear functions $\mathcal{A} \to \mathbb{C}$ 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 $C^\ast$-algebra of bounded linear operators on a Hilbert space. In other words, every abstract $C^*$-algebra may be made into a concrete $C^*$-algebra.
The Gelfand-Naimark-Segal construction (GNS construction) establishes a correspondence between cyclic $*$-representations of $C^*$-algebras and certain linear functionals (usually called states) on those same $C^*$-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 $C^*$-algebra over the complex numbers is that of complex-valued continuous functions from some compact Hausdorff topological space: there is an equivalence of categories $C^* CAlg \simeq$ Top${}_{cpt}$.
Accordingly one may think of the study of non-commutative $C^\ast$-algebras as non-commutative topology.
For $A$ and $B$ two $C^\ast$-algebras and $f : A \to B$ a star-algebra homomorphism the set-theoretic image $f(A) \subset B$ is a $C^\ast$-subalgebra of $B$, hence is also the image of $f$ in $C^\ast Alg$.
This is (KadisonRingrose, theorem 4.1.9).
There is a functor
to the category Poset of posets, which sends each $A \in C^\ast Alg$ to its poset of commutative subalgebras $\mathcal{C}(A)$ and sends each morphism $f : A \to B$ to the functor $\mathcal{C}(f) : \mathcal{C}(A) \to \mathcal{C}(B)$ which sends a commutative subalgebra $C \subset A$ to $f(C) \subset B$.
Many $C^\ast$-algebras arise as groupoid algebras of Lie groupoids. See at groupoid algebra - References - For smooth geometry
There is homotopy theory of $C^\ast$-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 $M_n(A)$ of matrices with coefficients in a $C^\ast$-algebra is again a $C^\ast$-algebra. In particular $M_n(\mathbb{C})$ is a $C^\ast$-algebra for all $n \in \mathbb{N}$.
For $A$ a $C^\ast$-algebra and for $X$ a locally compact Hausdorff topological space, the set of continuous functions $X \to A$ which vanish at infinity is again a $C^\ast$-algebra by extending all operations pointwise. (This algebra is unital precisely if $A$ is and if $X$ is a compact topological space.)
This algebra is denoted
If $A = \mathbb{C}$ then one usually just writes
This are the $C^\ast$-algebras to which the Gelfand duality theorem applies and which are the default algebras of observables in classical physics (for $X$ a phase space, cf. eg. Landsman (2017), §3).
The subalgebra $C_{00}(X) \subset C_0(X)$ of compactly supported among all vanishing at infinity-functions (Exp. ) is not in general itself a $C^\ast$-algebra, but is a very well-behaved ideal inside $C_0(X)$, cf. Amini (2004).
A uniformly hyperfinite algebra is in particular a $C^\ast$-algebra, by definition.
A von Neumann algebra is in particular a $C^\ast$-algebra, by definition.
Monographs:
Jacques Dixmier, Chapter 2 of: $C^\ast$-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, $C^\ast$-algebras and Operator Theory, Academic Press (1990) [doi:10.1016/C2009-0-22289-6]
Garth Warner, $C^\ast$-Algebras, EPrint Collection, University of Washington (2010) [hdl:1773/16302, pdf, pdf]
Ian Putnam, Lecture notes on $C^\ast$-algebras (2019) [pdf, pdf]
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]
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 $C^\ast$-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-$C^*$-algebras – hence of the poset of commutative subalgebras of a $C^*$-algebra – is in
General properties of the category of $C^\ast$-algebras are discussed in
Specifically pullback and pushout of $C^\ast$-algebras is discussed in
See also
The homotopy theory of $C^\ast$-algebras (a category of fibrant objects-structure on $C^\ast Alg$) is discussed in
For more along such lines see the references at KK-theory and E-theory.
Discussion of $C^\ast$-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.