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C-star-algebra in nLab

nLab C-star-algebra

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Contents

Context

Algebra

Group theory

Ring theory

Module theory

Gebras

Operator algebra

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

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Noncommutative geometry

Index theory

Contents

Definitions

Abstract C *C^\ast-algebras

Definition

A C *C^*-algebra is a Banach algebra (A,‖−‖)(A, {\|-\|}) over a topological field KK (often the field K≔ℂK \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.

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‖{\|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.

Remark

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

Remark

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

Definition

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.

Definition

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.

Remark

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.

Remark

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).

Properties

Category theoretic properties

C *C^*-algebras are monadic over sets. More precisely, the forgetful functor C *Alg→Set\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)≔{A∈𝒜|A *=A} Herm(\mathcal{A}) \;\coloneqq\; \big\{ A \,\in\, \mathcal{A} \,\big\vert\, A^\ast \,=\, A \big\}

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

A≥B⇔∃C∈𝒜A−B=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

A≥0⇔∃C∈𝒜A=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 real vector space of 𝒜\mathcal{A} that determines which linear functions 𝒜→ℂ\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 *CAlg≃C^* CAlg \simeq Top cpt{}_{cpt}.

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

General

Proposition

For AA and BB two C *C^\ast-algebras and f:A→Bf : 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).

Corollary

There is a functor

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

to the category Poset of posets, which sends each A∈C *AlgA \in C^\ast Alg to its poset of commutative subalgebras 𝒞(A)\mathcal{C}(A) and sends each morphism f:A→Bf : 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 C⊂AC \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.

Examples

Example

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(ℂ)M_n(\mathbb{C}) is a C *C^\ast-algebra for all n∈ℕn \in \mathbb{N}.

Example

For AA a C *C^\ast-algebra and for XX a locally compact Hausdorff topological space, the set of continuous functions X→AX \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=ℂA = \mathbb{C} then one usually just writes

C 0(X)≔C 0(X,ℂ). 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).

Remark

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).

Example

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

Example

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

References

Monographs:

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 November 9, 2024 at 04:24:04. See the history of this page for a list of all contributions to it.

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