Prove that for in H,
Thus, N is a homomorphism from onto the positive real numbers.
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Also because a1*(a2*(b))=(a1*a2)*b we know that A_a1*A_a2=A_(a1*a2) for any two real number a1, a2 (in other words, a->A_a is a morphism, in fact an isomorphism because A_a is zero iff x=y=z=t=0 iff a=0).
(a) According to what we said it is enough to show that A_conj(ab)=A_b*A_a. But we know that A_conj(ab)=transpose(A_ab) and A_conj(a)=transpose(A_a) and A_conj(b)=transpose(A_b) so it is ...