nLab C-star-algebra




Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT

Noncommutative geometry

Index theory



Abstract C *C^\ast-algebras


A C *C^*-algebra is a Banach algebra (A,)(A, {\|-\|}) over a topological field KK (often the field KK \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 *C^*-identity

A *A=A *A {\|{A^* A}\|} = {\|{A^*}\|} \, {\|{A}\|}

or equivalently the B *B^*-identity

A *A=A 2. {\|{A^* A}\|} = {\|{A}\|^2} \,.

A homomorphism of C *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 *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{\|A^*\|} = {\|A\|}, which is key for the equivalence of the B *B^* and C *C^* identities). This follows easily from the B *B^*-identity, while it follows from the C *C^*-identity after some difficulty.


There are different concepts for the tensor product of C *C^*-algebras, see for example at spatial tensor product.


C *C^*-algebras equipped with the action of a group by automorphisms of the algebra are called C-star-systems .

Concrete C *C^\ast-algebras and C *C^\ast-representations


Given a complex Hilbert space HH, a concrete C *C^*-algebra on HH is a **-subalgebra? of the algebra of bounded operators on HH that is closed in the norm topology.


A representation of a C *C^*-algebra AA on a Hilbert space HH is a **-homomorphism from AA to the algebra of bounded operators on HH.


It is immediate that concrete C *C^*-algebras correspond precisely to faithful representations of abstract C *C^*-algebras. It is an important theorem that every C *C^*-algebra has a faithful representation; that is, every abstract C *C^*-algebra is isomorphic to a concrete C *C^*-algebra.


The original definition of the term ‘C *C^*-algebra’ was in fact the concrete notion; the ‘C’ stood for ‘closed’. Furthermore, the original term for the abstract notion was ‘B *B^*-algebra’ (where the ‘B’ stood for ‘Banach’). However, we now usually interpret ‘C *C^*-algebra’ abstractly. (Compare ‘W *W^*-algebra’ and ‘von Neumann algebra’.)

In \dagger-compact categories

The notion of C *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).


Category theoretic properties

C *C^*-algebras are monadic over sets. More precisely, the forgetful functor C *AlgSet\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.

Partial order and positive elements

The self-adjoint elements in a C *C^\ast-algebra 𝒜\mathcal{A}

Herm(𝒜){A𝒜|A *=A} Herm(\mathcal{A}) \;\coloneqq\; \big\{ A \,\in\, \mathcal{A} \,\big\vert\, A^\ast \,=\, A \big\}

form a partially ordered complex vector space by declaring an element AA to be “larger” than some BB if the difference is a normal operator

ABC𝒜AB=C *C. A \geq B \;\;\;\;\; \Leftrightarrow \;\;\;\;\; \underset{C \in \mathcal{A} }{\exists} \;\; A - B \,=\, C^\ast C \,.

In particular, the positive elements are exactly the normal operators

A0C𝒜A=C *C. A \geq 0 \;\;\;\;\; \Leftrightarrow \;\;\;\;\; \underset{C \in \mathcal{A} }{\exists} \;\; A \,=\, C^\ast C \,.

(It is (only) this partial order on the underlying complex vector space of 𝒜\mathcal{A} that determines which linear functions 𝒜\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.

Gelfand-Naimark theorem

The Gelfand-Naimark theorem says that every C*-algebra is isomorphic to a C *C^\ast-algebra of bounded linear operators on a Hilbert space. In other words, every abstract C *C^*-algebra may be made into a concrete C *C^*-algebra.

Gelfand-Naimark-Segal construction

The Gelfand-Naimark-Segal construction (GNS construction) establishes a correspondence between cyclic **-representations of C *C^*-algebras and certain linear functionals (usually called states) on those same C *C^*-algebras. The correspondence comes about from an explicit construction of the *-representation from one of the linear functionals (states).

Gelfand duality

Gelfand duality says that every (unital) commutative C *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 *CAlgC^* CAlg \simeq Top cpt{}_{cpt}.

Accordingly one may think of the study of non-commutative C *C^\ast-algebras as non-commutative topology.



For AA and BB two C *C^\ast-algebras and f:ABf : A \to B a star-algebra homomorphism the set-theoretic image f(A)Bf(A) \subset B is a C *C^\ast-subalgebra of BB, hence is also the image of ff in C *AlgC^\ast Alg.

This is (KadisonRingrose, theorem 4.1.9).


There is a functor

𝒞:C *AlgPoset \mathcal{C} : C^\ast Alg \to Poset

to the category Poset of posets, which sends each AC *AlgA \in C^\ast Alg to its poset of commutative subalgebras 𝒞(A)\mathcal{C}(A) and sends each morphism f:ABf : A \to B to the functor 𝒞(f):𝒞(A)𝒞(B)\mathcal{C}(f) : \mathcal{C}(A) \to \mathcal{C}(B) which sends a commutative subalgebra CAC \subset A to f(C)Bf(C) \subset B.

Construction as groupoid convolution algebras

Many C *C^\ast-algebras arise as groupoid algebras of Lie groupoids. See at groupoid algebra - References - For smooth geometry

Homotopy theory

There is homotopy theory of C *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)M_n(A) of matrices with coefficients in a C *C^\ast-algebra is again a C *C^\ast-algebra. In particular M n()M_n(\mathbb{C}) is a C *C^\ast-algebra for all nn \in \mathbb{N}.


For AA a C *C^\ast-algebra and for XX a locally compact Hausdorff topological space, the set of continuous functions XAX \to A which vanish at infinity is again a C *C^\ast-algebra by extending all operations pointwise. (This algebra is unital precisely if AA is and if XX is a compact topological space.)

This algebra is denoted

C 0(X,A)C *Alg. C_0(X,A) \in C^\ast Alg \,.

If A=A = \mathbb{C} then one usually just writes

C 0(X)C 0(X,). C_0(X) \coloneqq C_0(X, \mathbb{C}) \,.

This are the C *C^\ast-algebras to which the Gelfand duality theorem applies and which are the default algebras of observables in classical physics (for XX a phase space, cf. eg. Landsman (2017), §3).


The subalgebra C 00(X)C 0(X)C_{00}(X) \subset C_0(X) of compactly supported among all vanishing at infinity-functions (Exp. ) is not in general itself a C *C^\ast-algebra, but is a very well-behaved ideal inside C 0(X)C_0(X), cf. Amini (2004).


A uniformly hyperfinite algebra is in particular a C *C^\ast-algebra, by definition.


A von Neumann algebra is in particular a C *C^\ast-algebra, by definition.



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

  • Ivo Dell’Ambrogio, Categories of C *C^\ast-algebras (pdf)

See also:

For operator algebra-theory see there and see

On category-theoretic properties:

  • J Wick Pelletier, J Rosicky, On the equational theory of C *C^*-algebras, Algebra Universalis 30 (1993) 275-284

A characterizations of injections of commutative sub-C *C^*-algebras – hence of the poset of commutative subalgebras of a C *C^*-algebra – is in

General properties of the category of C *C^\ast-algebras are discussed in

Specifically pullback and pushout of C *C^\ast-algebras is discussed in

  • Gerd Petersen, Pullback and pushout constructions in C *C^\ast-algebra theory (pdf)

See also

The homotopy theory of C *C^\ast-algebras (a category of fibrant objects-structure on C *AlgC^\ast Alg) is discussed in

  • Otgonbayar Uuye, Homotopy theory for C *C^\ast-algebras (arXiv:1011.2926)

For more along such lines see the references at KK-theory and E-theory.

Discussion of C *C^\ast-algebras as algebras of observables in quantum physics/quantum probability theory:

Last revised on March 19, 2024 at 07:04:28. See the history of this page for a list of all contributions to it.