We know all infinite dimensional Hilbert Spaces are unitarily equivalent. It should follow therefore that if I have an unitary representation of say Lorentz or Poincare group on one infinite dimensional Hilbert Space A I should be able to induce a representation on any other infinite dim Hilbert Space B using the unitary map between A and B. So for instance the Simple Harmonic Oscillator Hilbert Space will carry a representation of the Poincare group. Is this statement correct?
If so, suppose I have Hilbert Space which carries an irreducible representation of Poincare Group (like the space of positive frequency solutions to Klein Gordon equation in Minkowski space). Suppose also that it admits a tensor decomposition into some factor Hilbert Spaces. Then by the argument above, each factor Hilbert Space carries a representation of the Poincare Group (with possibly no geometric interpretation). But this would contradict the statement that the original representation was irreducible. Then what gives?