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Theorem - Let
 be  -algebras and
 a *-homomorphism. Then  is bounded and
 (where  is the norm of  seen as a linear operator between the spaces
 and
 ).
For this reason it is often said that homomorphisms between  -algebras are automatically continuous.
Corollary - A *-isomorphism between  -algebras is an isometric isomorphism.
Proof of Theorem : Let us first suppose that
 and
 have identity elements, both denoted by  .
We denote by  and
 the spectrum and the spectral radius of an element
 or
 .
Let
 and
 . If
 is invertible in
 , then
 is invertible in
 . Thus,
Hence
 for every
 . Therefore, by the result from this entry,
We conclude that  is bounded and
 .
If
 or
 do not have identity elements, we can consider their minimal unitizations, and the result follows from the above argument. 
Proof of Corollary : This follows from the fact that  is also a *-homomorphism and therefore
 for every
 . 
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