# Modules of algebras with idempotents and the Stone-von Neumann theorem

The Stone-von Neumann theorem tells us that all unitary irreducible representations of the integrated/exponentiated/Weyl form of the canonical commutation relations (CCR) algebra in finite dimensions are unitarily equivalent. (More generally, one may think of unitary irreducible central representations of the finite-dimensional Heisenberg group, that is, the ones where the center acts by a non-zero non-trivial scalar.)

For the sake of simplicity consider the one-dimensional case, i.e. $2\cdot 1$ parameters. See below for the definitions.

There are various proofs of this unicity result but von Neumann gave a clever argument by noticing that the linear combination $$P_{\varphi}=\int\mathrm{d}\sigma\,\mathrm{d}\tau\, \varphi(\sigma,\tau)S(\sigma,\tau)\,,\quad \varphi(\sigma,\tau)=\frac{1}{2\pi}e^{-\frac{\sigma^2+\tau^2}{4}}\,,\quad \sigma\,,\tau \in\mathbb{R}\,,$$ is a self-adjoint projection operator in any representation of the integrated CCR algebra. (In fact, it satisfies more: $P_\varphi S(\sigma,\tau)P_\varphi=e^{-\frac{\sigma^2+\tau^2}{4}}P_\varphi$, giving you projection for $\sigma=\tau=0$.) Irreducibility requires $P_\varphi$ to be a rank-one projection. Then, essentially, the map sending a unit vector in the image of $P_\varphi$ on one representation to a unit vector on the image of $P'_\varphi$ on any other representation extends linearly to a unitary intertwiner giving the desired equivalence. von Neumann's original article here. See also here for a selective history.

The existence of such non-trivial idempotent elements on an algebra seems to me, and hence the question, non-trivial. My question is then:

Question: What's the special structure behind the integrated CCR algebra (or the Heisenberg group) that makes its elements cohere so strongly to such an idempotent?

From a representation-theoretic point of view this structure is very constraining since it seems to be enough to characterize all of the algebra's simple modules.

I don't know much about systems of imprimitivity and Morita equivalence but perhaps the "right" setting to explore and understand such structures is in this context (or not!). Any specific pointers to the literature are welcome.

### Definitions

The CCR algebra means differents things to different people according to the level of depth one wants to work with. I'm thinking of the exponentiated CCR algebra as the algebra over $\mathbb{C}$ generated by the elements $S(\sigma,\tau)$ with $\sigma,\tau\in\mathbb{R}$, composing according to $$S(\sigma,\tau)S(\sigma',\tau')=e^{i(\sigma\tau'-\sigma'\tau)/2}S(\sigma+\sigma',\tau+\tau')\,.$$ (Note that $S(0,0)=\mathbb{1}$.) These relations may be obtained from the more familiar operators $U_t,V_s$, the exponentiated position and momentum operators, via $$S(\sigma,\tau)=e^{i\sigma\tau/2}U_\tau V_\sigma=e^{-i\sigma\tau/2}V_\sigma U_\tau\,.$$ In the usual Schroedinger representation $U_\tau,V_\sigma$ act on $f\in L^2(\mathbb{R})$ as multiplication $(U_\tau f)(x)=e^{i\tau x}f(x)$ and translation $(V_\sigma f)(x)=f(x+\sigma)$, respectively.

Slightly related mathoverflow questions: here, here, and here.

• I don't want to mess with your tags, but we have [heisenberg-groups] and [weyl-algebra] tags, which may be appropriate in place of some of the less specific ones you've got. Feb 21, 2018 at 14:29
• Indeed! Wasn't aware of them. Thank you! Feb 21, 2018 at 14:54