Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2502.00387 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929693600514048 |
|---|---|
| author | Bekka, Bachir |
| author_facet | Bekka, Bachir |
| contents | Let $R$ be a (not necessary commutative) ring with unit, $d\geq 1$ an integer, and $λ$ a unitary character of the additive group $(R,+).$ A pair $(U,V)$ of unitary representations $U$ and $V$ of $R^d$ on a Hilbert space $\mathcal{H}$ is said to satisfy the canonical commutation relations (relative to $λ$) if $U(a) V(b)= λ(a\cdot b)V(b) U(a)$ for all $a=(a_1, \dots, a_d), b= (b_1, \dots, b_d)\in R^d$, where $a\cdot b= \sum_{k=1}^d a_k b_k.$ We give a new and quick proof of the classical Stone von Neumann Theorem about the essential uniqueness of such a pair in the case where $R$ is a local field (e.g. $R= \mathbf{R}$). Our methods allow us to give the following extension of this result to a general locally compact ring $R$. For a unitary representation $U$ of $R^d$ on a Hilbert space $\mathcal{H}, $ define the inflation $U^{(\infty)}$ of $U$ as the (countably) infinite multiple of $U$ on $\mathcal{H}^{(\infty)}=\oplus_{i\in \mathbf{N}} \mathcal{H}$. Let $(U_1, V_1), (U_2, V_2)$ be two pairs of unitary representations of $R^d$ on corresponding Hilbert spaces $\mathcal{H}_1, \mathcal{H}_2$ satisfying the canonical commutation relations (relative to $λ$). Provided that $λ$ satisfies a mild faithful condition, we show that the inflations $(U_1^{(\infty)}, V_1^{(\infty)}), (U_2^{(\infty)}, V_2^{(\infty)})$ are approximately equivalent, that is, there exists a sequence $(Φ_n)_n$ of unitary isomorphisms $Φ_n: \mathcal{H}_1^{(\infty)}\to \mathcal{H}_2^{(\infty)}$ such that $\lim_{n} \Vert U_2^{(\infty)}(a) - Φ_n U_1^{(\infty)}(a) Φ_n^{*}\Vert=0$ and $\lim_{n} \Vert V_2^{(\infty)}(b) - Φ_n V_1^{(\infty)}(b) Φ_n^{*}\Vert=0,$ uniformly on compact subsets of $R^d.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canonical Commutation Relations: A quick proof of the Stone-von Neumann theorem and an extension to general rings Bekka, Bachir Representation Theory Mathematical Physics Group Theory 22D10, 43A25, 22B0 Let $R$ be a (not necessary commutative) ring with unit, $d\geq 1$ an integer, and $λ$ a unitary character of the additive group $(R,+).$ A pair $(U,V)$ of unitary representations $U$ and $V$ of $R^d$ on a Hilbert space $\mathcal{H}$ is said to satisfy the canonical commutation relations (relative to $λ$) if $U(a) V(b)= λ(a\cdot b)V(b) U(a)$ for all $a=(a_1, \dots, a_d), b= (b_1, \dots, b_d)\in R^d$, where $a\cdot b= \sum_{k=1}^d a_k b_k.$ We give a new and quick proof of the classical Stone von Neumann Theorem about the essential uniqueness of such a pair in the case where $R$ is a local field (e.g. $R= \mathbf{R}$). Our methods allow us to give the following extension of this result to a general locally compact ring $R$. For a unitary representation $U$ of $R^d$ on a Hilbert space $\mathcal{H}, $ define the inflation $U^{(\infty)}$ of $U$ as the (countably) infinite multiple of $U$ on $\mathcal{H}^{(\infty)}=\oplus_{i\in \mathbf{N}} \mathcal{H}$. Let $(U_1, V_1), (U_2, V_2)$ be two pairs of unitary representations of $R^d$ on corresponding Hilbert spaces $\mathcal{H}_1, \mathcal{H}_2$ satisfying the canonical commutation relations (relative to $λ$). Provided that $λ$ satisfies a mild faithful condition, we show that the inflations $(U_1^{(\infty)}, V_1^{(\infty)}), (U_2^{(\infty)}, V_2^{(\infty)})$ are approximately equivalent, that is, there exists a sequence $(Φ_n)_n$ of unitary isomorphisms $Φ_n: \mathcal{H}_1^{(\infty)}\to \mathcal{H}_2^{(\infty)}$ such that $\lim_{n} \Vert U_2^{(\infty)}(a) - Φ_n U_1^{(\infty)}(a) Φ_n^{*}\Vert=0$ and $\lim_{n} \Vert V_2^{(\infty)}(b) - Φ_n V_1^{(\infty)}(b) Φ_n^{*}\Vert=0,$ uniformly on compact subsets of $R^d.$ |
| title | Canonical Commutation Relations: A quick proof of the Stone-von Neumann theorem and an extension to general rings |
| topic | Representation Theory Mathematical Physics Group Theory 22D10, 43A25, 22B0 |
| url | https://arxiv.org/abs/2502.00387 |