Infinite matrices may violate the associative law
Ofir E. Alon, Nimrod Moiseyev, Asher Peres · Journal of Physics A Mathematical and General · 1995
The momentum operator for a particle in a box is represented by an infinite-order Hermitian matrix P. Its square P 2 is well-defined (and diagonal), but its cube P 3 is ill-defined, because P P 2 not=P 2 P. Truncating these matrices to a finite order restores the associative law, but leads to other curious results.