The universal Weyl C*-algebra

Assume Countable Choice. Let $(V,\omega )$ be a finite-dimensional real symplectic vector space and let $\lambda \in \mathbb{R}$. Regard the additive group underlying $V$ as discrete for this construction, and define $m_{\lambda }(v,w):=e^{-i\lambda \omega (v,w)\mathrm{/}2}\mathrm{.}$ Let $A_0(V,\omega ,\lambda )={\mathbb{C}}^{(V)}$ be the complex vector space of finitely supported functions on $V$, with basis $({\delta }_v{)}_{v\in V}$ and unit ${\delta }_0$. Define ${\delta }_v{\delta }_w=m_{\lambda }(v,w){\delta }_{v+w},{\delta }_v^{\ast }={\delta }_{-v},$ and extend multiplication bilinearly and the involution conjugate-linearly. Here a linear functional $\phi :A_0\to \mathbb{C}$ is positive unital if $\phi (a^{\ast }a)\ge 0$ for every $a\in A_0$ and $\phi ({\delta }_0)=1$. Put $N_{\phi }=\{a\in A_0:\phi (a^{\ast }a)=0\}$, equip $A_0\mathrm{/}{N}_{\phi }$ with $⟨[a],[b]{⟩}_{\phi }:=\phi (b^{\ast }a),$ and let ${\pi }_{\phi }$ be left multiplication on the Hilbert completion of this quotient. The lemma The universal Weyl norm is finite and has the C*-completion property ↗ verifies that these quotients and representations are well-defined. Define the universal C*-seminorm and completion by $\mathrm{\parallel }a{\mathrm{\parallel }}_u:={\sup }_{\phi }\mathrm{\parallel }{\pi }_{\phi }(a)\mathrm{\parallel },C_{\mathrm{W}}^{\ast }(V,\omega ,\lambda ):={\overline{A_0\mathrm{/}\ker \mathrm{\parallel }\cdot {\mathrm{\parallel }}_u}}^{\mathrm{\parallel }\cdot {\mathrm{\parallel }}_u}\mathrm{.}$ The lemma The universal Weyl norm is finite and has the C*-completion property ↗ proves that the twisted algebra and GNS representations are well-defined, the seminorm is finite and nonzero, and the completion is universal for unitary generators with the displayed Weyl relations. Countable Choice is included through the Hilbert completions in that construction.

The universal algebra is built from bounded unitary symbols and treats the underlying additive group as discrete. It does not impose strong continuity on $v\mapsto \pi ({\delta }_v)$; regularity is a property of a representation. It is distinct from the polynomial Weyl quotient of Polynomial Weyl algebra and its central parameter, whose generators are polynomial and whose definition has no C*-norm or topology. The local descriptions here state the distinction; the PA-07 item is not treated as an accepted dependency.