Operatoralgebra

Operator algebras are studied in the mathematical branch of functional analysis. These are generalizations of Matrizenalgebren linear algebra.

Introduction

Are normed spaces and and steady, linear operators, as well as their composition is a continuous, linear operator, and applies to the operator standards. Therefore, the space of continuous linear operators from into with the composition as multiplication in a normed ring, which when fully even is a Banach algebra.

Current algebras and their subalgebras are called operator algebras, the case that a Hilbert space is, is particularly intensively studied. Some authors understand by the term operator algebra only this Hilbert space case, this is especially true for older literature. Thus, the fundamental 1936-1943 published works by Francis J. Murray and John von Neumann, the title on wear rings of operators and treat algebras which is now called von Neumann algebras.

Banach algebras as operator algebras

Each normalized algebra can be represented as operator algebra. The so-called left regular representation of which assigns to each element of the operator, which is an isometric homomorphism, if an element has one. If no one element is present, so you adjungiere one.

What homomorphisms exist from a Banach algebra into an operator algebra, is studied in representation theory. A special interest is directed towards representations on Hilbert spaces, ie, homomorphisms into the operator algebra on a Hilbert space, which leads to the concepts of von Neumann algebra and C * - algebra.

Importance

Operator algebras on Banach spaces, especially on Hilbert spaces, allow the introduction of additional topologies, such as the strong or weak operator topology, with just the latter are of particular importance because of the compactness of the unit sphere.

Another structural element of operator algebras in which is not present in any Banach algebras are invariant subspaces, ie subspaces for which applies to any or all operators of the algebra. Especially in the Hilbert space case, the orthogonal projections onto invariant subspaces are generally not included in the operator algebra but in the commutant.

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

Operator algebras, in addition to the operator norm carry further standards and respect this be complete. In Hilbert spaces the adjunction of operators is added as an additional structural element and may define an involution on the considered algebras. Here may be mentioned especially the shadow classes, where the special case of trace class operators occur in the form of mixed states in the mathematical formulation of quantum mechanics.