nLab C-star-algebra

Definitions

Abstract $C^\ast$-algebras

Definition

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

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

or equivalently the $B^*$-identity

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

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.

Remark

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.

Remark

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

Remark

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

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

Definition

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.

Definition

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

Remark

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.

Remark

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’.)

In $\dagger$-compact categories

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).

Properties

Category theoretic properties

$C^*$-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.

Partial order and positive elements

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

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

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

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

In particular, the positive elements are exactly the normal operators

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

(It is (only) this partial order on the underlying complex vector space of $\mathcal{A}$ 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.

Gelfand-Naimark theorem

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.

Gelfand-Naimark-Segal construction

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

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.

General

Proposition

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).

Corollary

There is a functor

$$\mathcal{C} : C^\ast Alg \to Poset$$

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 $f(C) \subset B$.

Construction as groupoid convolution algebras

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

Homotopy theory

There is homotopy theory of $C^\ast$-algebras, being a non-commutative generalization of that of Top. (e.g. Uuye 12). For more see at homotopical structure on C*-algebras.

Examples

Example

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 $n \in \mathbb{N}$.

Example

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

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

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

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

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).

Remark

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).

Example

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

Example

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

Related concepts

• commutative C-star-algebra, Gelfand duality, noncommutative topology

• separable C*-algebra, homogeneous C*-algebra

• nuclear C*-algebra

• unitisation of C*-algebras

• von Neumann algebra, enveloping von Neumann algebra

• JB-algebra, JLB-algebra

• dense subalgebra, F-star-algebra

• multiplier algebra

• graded C*-algebra

• tensor product of C*-algebras

• crossed product C*-algebra

• Cuntz algebra

• Poincaré duality C*-algebra

• Hilbert C*-module, Hilbert C*-bimodule, amplimorphism

• C*-category

• C*-coalgebra, Hopf C*-algebra

• continuous field of C*-algebras

• homotopical structure on C*-algebras

  • asymptotic C*-homomorphism, l.m.c.-C*-algebra

  • KK-theory, E-theory

• AQFT

  • state on an operator algebra

References

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

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

See also:

• Wikipedia, $C^\ast$-algebra

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:

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

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

• Chris Heunen, Characterizations of categories of commutative $C^*$-algebras (arXiv:1106.5942)

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

• Ralf Meyer, Categorical aspects of bivariant K-theory, (arXiv:math/0702145)

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

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

See also

• Massoud Amini, Locally Compact Pro-$C^\ast$-Algebras, Canadian Journal of Mathematics 56 1 (2004) 3-22 [arXiv:math/0205253, doi:10.4153/CJM-2004-001-6]

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

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

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:

• Klaas Landsman, Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open (2017) [doi:10.1007/978-3-319-51777-3, pdf]