I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar.
Can anybody help, please?
One side is obvious, so our concern is the other side. I tried to use Polar Decomposition.. If M,N are the two operators then given is that. $ N=SMS^{-1} $ then we have to find U which is unitary such that $N=U M U^{-1} $. I tried to write $ S=UR $ where U is unitary and R is positive. Then $ N=URMR^{-1}U^{-1} $. I want to show $ RMR^{-1}=M $ where it is given that M and N are normal and $R=(SS^*)^{1/2} $ . Could not proceed further.
You can write the similarity as $NS=SM $. As $N $ and $M $ are normal, the Fuglede-Putnam theorem guarantees that $N^*S=SM^*$. Taking adjoints, $S^*N=MS^*$. Then $$ S^*SM=S^*NS=MS^*S. $$ Using this identity repeteadly, $p (S^*S)M=Mp (S^*S ) $ for all polynomials; taking limits, $f (S^*S)M=Mf (S^*S) $ for all continuous functions $f $. In particular, if $S=UR $ is the polar decomposition,$$ RM=MR. $$ Note that $U $ is also in the C $^*$-algebra because $S $ is invertible. Now $$N=SMS^{-1}=URMR^{-1}U^*=UMU^*. $$