injective C*-algebra homomorphism is isometric

Theorem - Let 𝒜 and be C*-algebras ( and Φ:𝒜 an injectivePlanetmathPlanetmath *-homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Then Φ(x)=x and σ(Φ(x))=σ(x) for every x𝒜, where σ(y) denotes the spectrum of the element y.

Proof: It suffices to prove the result for unital C*-algebras, since the general case follows directly by considering the minimal unitizations of 𝒜 and . So we assume that 𝒜 and are unital and we will denote their identity elementsMathworldPlanetmath by e, being clear from context which one is being used.

Let us first prove the second part of the theorem for normal elementsMathworldPlanetmath x𝒜. It is clear that σ(Φ(x))σ(x) since if x-λe invertiblePlanetmathPlanetmath for some λ𝒞, then so is Φ(x)-λe=Φ(x-λe). Suppose the inclusion is strict, then there is a non-zero function fC(σ(x)) whose restrictionPlanetmathPlanetmath to σ(Φ(x)) is zero (here C(σ(x)) denotes the C*-algebra of continuous functionsPlanetmathPlanetmath σ(x)). Thus we have, by the continuous functional calculus, that f(x)0 and also that


by the continuous functional calculus and the result on this entry ( Thus, we conclude that Φ is not injective and which is a contradictionMathworldPlanetmathPlanetmath. Hence we must have σ(Φ(x))=σ(x).

Let Rσ(z) denote the spectral radius of the element z. From the norm and spectral radius relationMathworldPlanetmathPlanetmath in C*-algebras ( we know that, for an arbitrary element x𝒜, we have that


Since the element x*x is normal, from the preceding paragraph it follows that Rσ(x*x)=Rσ(Φ(x*x)), and hence we conclude that


i.e. Φ(x)=x.

Since Φ is isometric, Φ(𝒜) is closed *-subalgebraMathworldPlanetmath of , i.e. Φ(𝒜) is a C*-subalgebra of , and it is isomorphic to 𝒜. Using the spectral invariance theorem we conclude that σ(x)=σ(Φ(x)) for every x𝒜.

Title injective C*-algebra homomorphism is isometric
Canonical name InjectiveCalgebraHomomorphismIsIsometric
Date of creation 2013-03-22 18:00:35
Last modified on 2013-03-22 18:00:35
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Theorem
Classification msc 46L05