C*-algebras have approximate identities

In this entry has three different meanings:

  1. 1.

    - The ordering of self-adjoint elements (http://planetmath.org/OrderingOfSelfAdjoints) of a given C*-algebra (http://planetmath.org/CAlgebra).

  2. 2.

    - The usual order (http://planetmath.org/PartialOrder) in .

  3. 3.

    - The of a directed set taken as the domain of a given net.

It will be clear from the context which one is being used.

Theorem - Every C*-algebra has an approximate identity (eλ)λΛ. Moreover, the approximate identity (eλ)λΛ can be chosen to the following :

  • 0eλλΛ

  • eλ1λΛ

  • λμeλeμ, i.e. (eλ)λΛ is increasing.

For separablePlanetmathPlanetmath (http://planetmath.org/Separable) C*-algebras the approximate identity can be chosen as an increasing sequence 0e1e2 of norm-one elements.

Title C*-algebras have approximate identities
Canonical name CalgebrasHaveApproximateIdentities
Date of creation 2013-03-22 17:30:40
Last modified on 2013-03-22 17:30:40
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Theorem
Classification msc 46L05