Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 10 / 3 / a / Solution

A unitary family with the required Weyl relations isTranslations preserve the $L^2$ norm and the multiplier has modulus one. Multiplication of two such operators leaves the phase $x_1{y}_2-y_1{x}_2$ after extracting $W(x_1+x_2,y_1+y_2)$, proving the relation. Translation and modulation are strongly continuous, first on compactly supported smooth functions by direct estimates, then on all of $L^2$ by density and unitarity.

If a bounded operator $A$ commutes with all $W$, it commutes with translations and every modulation $e^{2iyt}$. The spectral theorem for the multiplication coordinate therefore makes it commute with all bounded multiplication operators and interval projections. One can see its form directly: on a finite interval $I$, set $h_I=A{1}_I$. For bounded $\varphi$ supported in $I$, commutation gives $A\varphi =\varphi h_I$. Testing characteristic functions shows $|h_I|\le \Vert A\Vert$ almost everywhere. On increasing intervals these functions agree on overlaps and yield a single $h\in L^{\infty }$ with $A=M_h$.

Commutation with translations says $h(t+x)=h(t)$ almost everywhere for every $x$. Convolving with a smooth approximate identity produces smooth functions invariant under every translation, hence constants. Passing to the local $L^1$ limit makes $h$ constant almost everywhere. Therefore $A$ is a scalar operator, and the unitary family acts irreducibly.