Operator algebra

Operator algebras are studied in the mathematical sub-area of functional analysis . These are generalizations of the matrix algebras of linear algebra .

introduction

If normalized spaces and and are continuous , linear operators , then their composition is also a continuous, linear operator , and the operator norms apply . Therefore the space of the continuous, linear operators of in itself with the composition as multiplication becomes a normalized algebra , which is even a Banach algebra when complete . ${\displaystyle E,F,G}$ ${\displaystyle A:E\rightarrow F}$ ${\displaystyle B:F\rightarrow G}$ ${\displaystyle B\circ A:E\rightarrow G}$ ${\displaystyle \|B\circ A\|\leq \|B\|\cdot \|A\|}$ ${\displaystyle L(E)}$ ${\displaystyle E}$ ${\displaystyle E}$

These algebras and their sub-algebras are called operator algebras, with the case that a Hilbert space is being examined particularly intensively. Some authors understand the term operator algebra only to mean this Hilbert space case, this is especially true for older literature. The fundamental works by Francis J. Murray and John von Neumann , published from 1936 to 1943, are entitled On rings of operators and deal with algebras that are now called Von Neumann algebras . ${\displaystyle E}$

Banach algebras as operator algebras

Any normalized algebra can be represented as an operator algebra . The so-called left - regular representation of , which assigns the operator to each element , where , is an isometric homomorphism if it has a one element . If there is no unity, one should adjoint one. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle A}$ ${\displaystyle \ell _{A}\in L({\mathcal {A}})}$ ${\displaystyle \ell _{A}(B):=AB}$ ${\displaystyle {\mathcal {A}}}$

Which homomorphisms exist from a Banach algebra to an operator algebra is examined in representation theory. A special interest applies to representations on Hilbert spaces , that is, homomorphisms into the operator algebra over a Hilbert space, which leads to the terms Von Neumann algebra and C * algebra .

meaning

Operator algebras over Banach spaces, especially over Hilbert spaces, allow the introduction of additional topologies such as the strong or weak operator topology , the latter being of particular importance due to the compactness of the unit sphere .

Another structural element of operator algebras , which is not present in any Banach algebras, are invariant subspaces , that is, subspaces for which applies to individual or all operators of the algebra. Especially in the Hilbert space case, the orthogonal projections onto invariant subspaces are generally not contained in the operator algebra, but in its commutant . ${\displaystyle L(E)}$ ${\displaystyle U\subset E}$ ${\displaystyle A(U)\subset U}$ ${\displaystyle A}$

The unrestricted operators on a Hilbert space that are important for quantum mechanics do not form algebra, but can be related to operator algebras. Furthermore, because of the underlying space, one can speak of eigenvectors , which represent the states in quantum mechanics .

In addition to the operator norm, operator algebras can carry further norms and be complete with regard to this. On Hilbert spaces, the adjunction of operators is added as an additional structural element and can define an involution on the algebras under consideration. The shadow classes deserve special mention here, with the special case of trace class operators in the form of mixed states in the mathematical formulation of quantum mechanics.

Web links

• Higson, Roe: Operator algebras

Individual evidence

1. ^ FJ Murray, J. von Neumann: On rings of operators. Ann. of Math. (2), Volume 37, 1936, pages 116-229.
2. ^ FJ Murray, J. von Neumann: On rings of operators II. Trans. Amer. Math. Soc., Volume 41, 1937, pages 208-248
3. ^ FJ Murray, J. von Neumann: On rings of operators IV. Ann. of Math. (2), Vol. 44, 1943, pages 716-808.
4. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-5400-6386-2 , Chapter III, Representation Theory
5. ^ Jacques Dixmier : Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann , Gauthier-Villars, 1957 ( ISBN 2-87647-012-8 )
6. ^ Jacques Dixmier: Les C * -algèbres et leurs représentations , Gauthier-Villars, 1969 ( ISBN 2-87647-013-6 )
7. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , Volume I, 1983, ISBN 0-1239-3301-3 , Theorem 5.1.3
8. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras , Volume I, 1983, ISBN 0-1239-3301-3 , chapter 5.6
9. ^ Robert Schatten : Norm Ideals of Completely Continuous Operators. Results of mathematics and its border areas, 2nd part, ISBN 3-540-04806-5