Let V be the space of complex n×n matrices equipped with the inner product (A∣B)=tr(AB∗). If B is an element of V, let LB, RB, and TB denote the linear operators on V defined by
(a) LB(A)=BA.
(b) RB(A)=AB.
(c) TB(A)=BA−AB.
Consider the three families of operators obtained by letting B vary over all diagonal matrices. Show that each of these families is a commutative self-adjoint algebra and find their spectral resolutions.