injective -algebra homomorphism is isometric
Proof: It suffices to prove the result for unital -algebras, since the general case follows directly by considering the minimal unitizations of and . So we assume that and are unital and we will denote their identity elements by , being clear from context which one is being used.
Let us first prove the second part of the theorem for normal elements . It is clear that since if invertible for some , then so is . Suppose the inclusion is strict, then there is a non-zero function whose restriction to is zero (here denotes the -algebra of continuous functions ). Thus we have, by the continuous functional calculus, that and also that
by the continuous functional calculus and the result on this entry (http://planetmath.org/CAlgebraHomomorphismsPreserveContinuousFunctionalCalculus). Thus, we conclude that is not injective and which is a contradiction. Hence we must have .
Let denote the spectral radius of the element . From the norm and spectral radius relation in -algebras (http://planetmath.org/NormAndSpectralRadiusInCAlgebras) we know that, for an arbitrary element , we have that
Since the element is normal, from the preceding paragraph it follows that , and hence we conclude that
|Title||injective -algebra homomorphism is isometric|
|Date of creation||2013-03-22 18:00:35|
|Last modified on||2013-03-22 18:00:35|
|Last modified by||asteroid (17536)|