proof of quotients in C*-algebras

Proof: We have that is self-adjointPlanetmathPlanetmath (, since it is a closed ideal of a C*-algebraPlanetmathPlanetmath ( (see this entry ( Hence, the involutionMathworldPlanetmath in 𝒜 induces a well-defined involution in 𝒜/ by (x+)*:=x*+.

Recall that, since is closed, the quotient norm is indeed a norm in 𝒜/ that makes 𝒜/ a Banach algebraMathworldPlanetmath (see this entry ( Thus we only have to prove the C* to prove that 𝒜/ is a C*-algebra.

Recall that C*-algebras have approximate identities ( Notice that itself is a C*-algebra and pick an approximate identity (eλ) in such that

  • each eλ is positivePlanetmathPlanetmath.

  • eλ1

We will only prove the case when 𝒜 has an identity elementMathworldPlanetmath e. For the non-unital case, one can consider 𝒜 as a C*-subalgebra of its minimal unitization and the same proof will still work.

Let q denote the quotient norm in 𝒜/. We claim that for every x𝒜:

x+q=limλx(e-eλ) (1)

We will prove the above equality as a lemma at the end of the entry. Assuming this result, it follows that for every a𝒜


Since each eλ is positive and eλ1 we know that its spectrum lies on the interval [0,1]. Hence e-eλ is also positive and its spectrum also lies on the interval [0,1]. Thus, e-eλ1. Therefore:


Since 𝒜/ is a Banach algebra, we also have x*x+qx+q2 and so


which proves that 𝒜/ is a C*-algebra.

We now prove equality (1) as a lemma.

Lemma - Suppose 𝒜 is a C*-algebra with identity element e. Let 𝒜 be a closed ideal and (eλ) be an approximate identity in such that each eλ is positive and eλ1. Then


for every x in 𝒜.

Proof: Since y(e-eλ)0 for every y it follows that

lim supx(e-eλ) = lim supx-xeλ-y+yeλ
= lim sup(x-y)(e-eλ)

Therefore, taking the infimum over all y we obtain:

lim supx(e-eλ)infyx-y=x+q

Also, since xeλ,

lim infx(e-eλ)infyx-y=x+q

and this proves the lemma.

Title proof of quotientsPlanetmathPlanetmath in C*-algebras
Canonical name ProofOfQuotientsInCalgebras
Date of creation 2013-03-22 17:41:56
Last modified on 2013-03-22 17:41:56
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Proof
Classification msc 46L05