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Theorem - Let
 be a *-homomorphism between the  -algebras
 and
 . Then  has closed image, i.e.
 is closed in
 .
Thus, the image
 is a  -subalgebra of
 .
Proof: The kernel of  ,
 , is a closed two-sided ideal of
 , since  is continuous (see this entry). Factoring threw the quotient  -algebra
 we obtain an injective *-homomorphism
 .
Injective *-homomorphisms between  -algebras are known to be isometric (see this entry), hence the image
 is closed in
 .
Since the images
 and
 coincide we conclude that
 is closed in
 . 
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