I've been into this for days and days and I still can't see why, given the definition of $L^\ast$ as $L^\ast =(L(a^\ast))^\ast$ we get that $(LM)^\ast =L^\ast M^\ast$. Where is my mistake:
$$(LM)^\ast (a) =(L(a^\ast )M(a^\ast ))^\ast=(M(a^\ast ))^\ast (L(a^\ast ))^\ast =M^\ast (a)L^\ast (a)$$
This gives us $(LM)^\ast=M^\ast L^\ast$ which completely ruins the whole construction of the algebra of centralizers. I would be very grateful for any help!
Multiplication of left centralizers is defined as composition $(LM)(a) = L(M(a))$, not pointwise multiplication $L(a)M(a)$. Notice that $L,M \colon A \to A$ are bounded linear maps.
Thus $(M^\ast(a))^\ast = M(a^\ast)$ yields $$ (L^\ast M^\ast)(a) = L^\ast(M^\ast(a)) = (L[(M^\ast(a))^\ast])^\ast = (L[M(a^\ast)])^\ast = ((LM)(a^\ast))^\ast= (LM)^\ast(a). $$