closed ideals in C*-algebras are self-adjoint

Theorem - Every closed ( two-sided idealMathworldPlanetmath ( of a C*-algebra ( 𝒜 is self-adjoint (, i.e.

if x then x*.

Proof : Let *:={a*:a}.

Since is closed and the involution mapping is continuous, it follows that * is also closed.

We claim that * is also a of 𝒜. To see this let a,b, x𝒜 and λ. Then

  • a*+λb*=(a+λ¯b)** since a+λ¯b

  • xa*=(ax*)** since ax*.

  • a*x=(x*a)** since x*a

Let :=*.

is a C*-subalgebra of 𝒜 (it is a norm-closed, involution-closed, subalgebra of 𝒜).

It is known that every C*-algebra has an approximate identity consisting of positive elementsMathworldPlanetmathPlanetmath with norm less than 1 (see this entry (

Let (eλ)λΛ be an approximate identity for with the above :

  1. 1.

    each eλ is positive (hence self-adjoint) and

  2. 2.


We now prove is self-adjoint:

Let a. We have that

a*-a*eλ2 = (a*-a*eλ)*(a*-a*eλ)
= (a-eλa)(a*-a*eλ)
= (aa*-aa*eλ)-eλ(aa*-aa*eλ)
= 2aa*-aa*eλ

Taking limits in both we obtain

limλa*-a*eλ2limλ 2aa*-aa*eλ=0

since aa**= and (eλ)λΛ is an approximate identity for .

As eλ we see that a*eλ.

We conclude from the limit above that a* is in the closure of . Therefore a*.

Hence, is self-adjoint.

Title closed ideals in C*-algebras are self-adjoint
Canonical name ClosedIdealsInCalgebrasAreSelfadjoint
Date of creation 2013-03-22 17:30:42
Last modified on 2013-03-22 17:30:42
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 9
Author asteroid (17536)
Entry type Theorem
Classification msc 46L05
Classification msc 46H10