-algebra homomorphisms are continuous
Theorem - Let be -algebras (http://planetmath.org/CAlgebra) and a *-homomorphism. Then is bounded (http://planetmath.org/ContinuousLinearMapping) and (where is the norm (http://planetmath.org/OperatorNorm) of seen as a linear operator between the spaces and ).
For this reason it is often said that homomorphisms between -algebras are automatically continuous (http://planetmath.org/ContinuousLinearMapping).
Corollary - A *-isomorphism between -algebras is an isometric isomorphism (http://planetmath.org/IsometricIsomorphism).
Proof of Theorem : Let us first suppose that and have identity elements, both denoted by .
Let and . If is invertible in , then is invertible in . Thus,
Hence for every . Therefore, by the result from this entry (http://planetmath.org/NormAndSpectralRadiusInCAlgebras),
We conclude that is and .
If or do not have identity elements, we can consider their minimal unitizations, and the result follows from the above .
Proof of Corollary : This follows from the fact that is also a *-homomorphism and therefore for every .
|Title||-algebra homomorphisms are continuous|
|Date of creation||2013-03-22 17:40:06|
|Last modified on||2013-03-22 17:40:06|
|Last modified by||asteroid (17536)|
|Synonym||automatic continuity of -homomorphisms|
|Synonym||homomorphisms of -algebras are continuous|
|Defines||automatically continuous homomorphism of –algebras|