Gelfand-Neumark theorem

The Gelfand-Neumark theorems (after Israel Gelfand and Mark Neumark ) and the GNS construction form the starting point of the mathematical theory of C * algebras . They combine abstractly defined C * algebras with concrete algebras of functions and operators.

The first examples of C * -algebras that can be given directly after the definition are the algebra of continuous functions on a locally compact Hausdorff space X , which vanish at infinity ( see C ₀ -function ), and the sub-C * algebras of wherein the algebra of the limited, linear operators on a Hilbert space H is. ${\displaystyle C_{0}(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle L(H)}$ ${\displaystyle L(H)}$

The Gelfand-Neumark theorems show that apart from isometric * isomorphism, these are all possible C * algebras. These results are astonishing, because the definition of the C * -algebras does not speak of locally compact Hausdorff spaces or of Hilbert spaces.

Gelfand-Neumark theorem, commutative case

If A is a commutative C * algebra, then there is a locally compact Hausdorff space X and an isometric * isomorphism between A and . ${\displaystyle C_{0}(X)}$

Construction of the locally compact Hausdorff area

X is the set of all * -homomorphisms different from the zero map . Each is defined by a figure . Finally, one can prove that the topology of point-wise convergence makes X a locally compact Hausdorff space and that there is an isometric * -isomorphism between A and . ${\displaystyle \chi \colon A\rightarrow \mathbb {C} }$ ${\displaystyle a\in A}$ ${\displaystyle {\tilde {a}}(\chi )=\chi (a)}$ ${\displaystyle {\tilde {a}}:X\rightarrow \mathbb {C} }$ ${\displaystyle a\rightarrow {\tilde {a}}}$ ${\displaystyle C_{0}(X)}$

Remarks

According to this theorem, an element of a commutative C * -algebra can be treated like a continuous function, which can be expanded to a so-called continuous functional calculus . So is z. B. the spectrum of an element is nothing more than the completion of the picture of the associated continuous function.

This theorem opens up a very fruitful interplay between algebraic properties of C * -algebras and topological properties of locally compact spaces. Is , one has the following equivalents among many others: ${\displaystyle A\cong C_{0}(X)}$

• A has one element . X is compact . ${\displaystyle \Leftrightarrow }$
• A is finitely generated. X is homeomorphic to a subset of a finite-dimensional vector space. ${\displaystyle \Leftrightarrow }$
• A is separable . X satisfies the second axiom of countability . ${\displaystyle \Leftrightarrow }$
• A has a countable approximation of the one X is σ-compact . ${\displaystyle \Leftrightarrow }$
• The one- point compactification of X corresponds to the adjunction of a single element .
• The Stone-Čech compactification corresponds to the transition to the multiplier algebra .

Topological terms are translated into algebraic properties of commutative C * algebras and then generalized to non-commutative C * algebras; this is often the starting point for further theories. For this reason, the theory of C * algebras is also called non-commutative topology.

Gelfand-Neumark theorem, general case

If A is a C * -algebra, then there is a Hilbert space H such that A is isometric * -isomorphic to a sub-C * -algebra of L (H).

Construction of the Hilbert space

Be a continuous linear functional with and for everyone . Such functional also called states of A . The state set . Then the formula defines a scalar product on the quotient space . The completion with regard to this scalar product is a Hilbert space . For each imaging can be a continuous linear operator on continue. Then one shows that the mapping explained in this way is a * -homomorphism. Finally, one constructs a Hilbert space of the desired type from the entirety of the Hilbert spaces obtained in this way. ${\displaystyle f:A\rightarrow \mathbb {C} }$ ${\displaystyle \|f\|=1}$ ${\displaystyle f(x^{*}x)\geq 0}$ ${\displaystyle x\in A}$ ${\displaystyle f}$ ${\displaystyle N_{f}:=\{x\in A:f(x^{*}x)=0\}}$ ${\displaystyle \langle x+N_{f},y+N_{f}\rangle =f(y^{*}x)}$ ${\displaystyle A/N_{f}}$ ${\displaystyle H_{f}}$ ${\displaystyle a\in A}$ ${\displaystyle x+N_{f}\mapsto ax+N_{f}}$ ${\displaystyle \pi _{f}(a)}$ ${\displaystyle H_{f}}$ ${\displaystyle \pi _{f}:A\rightarrow L(H_{f})}$ ${\displaystyle H_{f}}$

Remarks

An element of an abstractly defined C * -algebra can thus be treated like a bounded linear operator on a Hilbert space.

The construction of from f described above is called the GNS construction , where GNS stands for Gelfand , Neumark and Segal . ${\displaystyle \pi _{f}}$

Is called * -Homomorphismen the type also representations from A to H. According to the above set each C * algebra has a loyal (d. H. Injective) representation on a Hilbert space. A representation is said to be topologically irreducible if there is no real closed subspace U of H other than 0 for which applies to all . ${\displaystyle \pi :A\rightarrow L(H)}$ ${\displaystyle \pi (a)U\subset U}$ ${\displaystyle a\in A}$

Segal's theorem

If A is a C * -algebra, then the state space S (A) is convex and is an extreme point if and only if the representation is topologically irreducible. ${\displaystyle f\in S(A)}$ ${\displaystyle \pi _{f}:A\rightarrow L(H_{f})}$

Every irreducible representation of A is of the form for an extremal state f of A. ${\displaystyle \pi _{f}}$

further remarks

On this basis, a very extensive representation theory for C * algebras was developed. C * algebras can be further classified by the images of their irreducible representations. A C * -algebra is called liminal if the image of every irreducible representation coincides with the algebra of the compact operators . A C * -algebra is called postliminal if the image of every irreducible representation contains the algebra of the compact operators.

literature

• Jacques Dixmier : Les C * -algèbres et leurs représentations. 2. édition. Gauthier-Villars, Paris 1969 ( Cahiers Scientifiques 29, ISSN  0750-2265 ).
• IM Gelfand , MA Neumark : On the embedding of normed rings into the ring of operators in Hilbert space. In: Matematiceskij sbornik. = Recueil mathématique. 54 = NS 12, 1943, ISSN  0368-8666 , pp. 197-213, online (PDF; 1.88 MB) .
• Richard V. Kadison , John R. Ringrose : Fundamentals of the Theory of Operator Algebras. Volume 2: Advanced Theory. Academic Press, New York NY 1986, ISBN 0-12-393350-1 ( Pure and Applied Mathematics 100, 2).
• IE Segal : Irreducible Representations of Operator Algebras. In: Bulletin of the American Mathematical Society. 53, 1947, ISSN  0002-9904 , pp. 73-88.
• Dirk Werner : Functional Analysis. 5th expanded edition. Springer, Berlin et al. 2005, ISBN 3-540-21381-3 , p. 466ff. ( Springer textbook ).

Individual evidence

1. ↑ Chun-Yen Chou: Notes on the separability of C * -algebras , Taiwanese Journal of Mathematics, Volume 16 (2), 2012, pages 555-559
2. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , sentence 3.10.5