symmetric monoidal (∞,1)-category of spectra
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 homomorphisms.
$C^\ast$-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 J Wick Pelletier & J Rosicky, On the equational theory of $C^*$-algebras, Algebra Universalis 30:275-284, 1993.
See also operator algebras.
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.
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.
Textbook accounts:
Gerard Murphy, $C^\ast$-algebras and Operator Theory, Academic Press 1990 (doi:10.1016/C2009-0-22289-6)
Richard Kadison, John Ringrose, chapter 4 in volume 1 of: Fundamentals of the theory of operator algebras Academic Press, (1983)
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
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)
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
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.
Last revised on May 11, 2021 at 05:46:32. See the history of this page for a list of all contributions to it.