symmetric monoidal (∞,1)-category of spectra
AQFT and operator algebra
(geometry $\leftarrow$ Isbell duality $\to$ algebra)
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
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 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).
See also operator algebras.
The GNS construction shows that every abstract $C^*$-algebra over the complex numbers as a concrete $C^*$-algebra: a subalgebra of an algebra of bounded operators on some Hilbert space.
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 $C^* CAlg_{com} \simeq$ 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.
A standard textbook reference is chapter 4 in volume 1 of
See also the references at operator algebras.
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
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.