Editing Projective representation (section)

===Infinite-dimensional projective unitary representations: the Heisenberg case===
The results of the previous subsection do not hold in the infinite-dimensional case, simply because the trace of <math>\rho_*(X)</math> is typically not well defined. Indeed, the result fails: Consider, for example, the translations in position space and in momentum space for a quantum particle moving in <math>\mathbb R^n</math>, acting on the Hilbert space <math>L^2(\mathbb R^n)</math>.<ref>{{harvnb|Hall|2013}} Example 16.56</ref> These operators are defined as follows:
:<math>\begin{align}
  (T_a f)(x) &= f(x - a) \\
  (S_a f)(x) &= e^{iax}f(x),
\end{align}</math>

for all <math>a\in\mathbb R^n</math>. These operators are simply continuous versions of the operators <math>T_a</math> and <math>S_a</math> described in the "First example" section above. As in that section, we can then define a ''projective'' unitary representation <math>\rho</math> of <math>\mathbb R^{2n}</math>:
:<math>\rho(a, b) = [T_a S_b],</math>
because the operators commute up to a [[phase factor]]. But no choice of the phase factors will lead to an ordinary unitary representation, since translations in position do not commute with translations in momentum (and multiplying by a nonzero constant will not change this). These operators do, however, come from an ordinary unitary representation of the [[Heisenberg group]], which is a one-dimensional central extension of <math>\mathbb R^{2n}</math>.<ref>{{harvnb|Hall|2013}} Exercise 6 in Chapter 14</ref> (See also the [[Stone–von Neumann theorem]].)