equivalence of definitions of C*-algebra


In this entry, we will prove that the definitions of C* algebra given in the main entry are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Theorem 1.

A Banach algebraMathworldPlanetmath A with an antilinear involution * such that a2a*a for all aA is a C*-algebra.

Proof.

It follows from the productPlanetmathPlanetmath inequality abab that

a2a*aa*a.

Therefore, aa*. Putting a* for a, we also have a*a**=a. Thus, the involution is an isometry: a=a*. So now,

a2a*aa2.

Hence, a*a=a2. ∎

Theorem 2.

A Banach algebra A with an antilinear involution * such that a*a=a*a is a C*-algebra.

Title equivalence of definitions of C*-algebra
Canonical name EquivalenceOfDefinitionsOfCalgebra
Date of creation 2013-03-22 17:42:27
Last modified on 2013-03-22 17:42:27
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 4
Author rspuzio (6075)
Entry type TheoremMathworldPlanetmath
Classification msc 46L05
Related topic HomomorphismsOfCAlgebrasAreContinuous
Related topic CAlgebra