ordering of self-adjoints

Let 𝒜 be a C*-algebra (http://planetmath.org/CAlgebra). Let 𝒜+ denote the set of positive elementsMathworldPlanetmathPlanetmathPlanetmath of 𝒜 and 𝒜sa denote the set of self-adjoint elementsMathworldPlanetmath of 𝒜.

Since 𝒜+ is a proper convex cone (http://planetmath.org/Cone5) (see this entry (http://planetmath.org/PositiveElement3)), we can define a partial orderMathworldPlanetmath on the set 𝒜sa, by setting

ab if and only if b-a𝒜+, i.e. b-a is positive.

Theorem - The relationMathworldPlanetmathPlanetmath is a partial order relation on 𝒜sa. Moreover, turns 𝒜sa into an ordered topological vector space.

0.0.1 Properties:

  • abc*acc*bc for every c𝒜.

  • If a and b are invertible and ab, then b-1a-1.

  • If 𝒜 has an identity elementMathworldPlanetmath e, then -aeaae for every a𝒜sa.

  • -babab.

0.0.2 Remark:

The proof that is partial order makes no use of the self-adjointness . In fact, 𝒜 itself is an ordered topological vector space under the relation .

However, it turns out that this ordering relation provides its most usefulness when restricted to self-adjoint elements. For example, some of the above would not hold if we did not restrict to 𝒜sa.

Title ordering of self-adjoints
Canonical name OrderingOfSelfadjoints
Date of creation 2013-03-22 17:30:37
Last modified on 2013-03-22 17:30:37
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Theorem
Classification msc 46L05