The universal Weyl C*-algebra has nonregular representations

On basis vectors of the finite-support subspace of ${\mathrm{\ell }}^2(V)$ define $L_v{\delta }_x=m_{\lambda }(v,x){\delta }_{v+x}$. Since $\mathrm{\mid }{m}_{\lambda }(v,x)\mathrm{\mid }=1$ and $x\mapsto v+x$ is a bijection of $V$, $L_v$ preserves the inner product and maps finite-support vectors to finite-support vectors. It extends to ${\mathrm{\ell }}^2(V)$ by $L_v[(a_n)]=[(L_v{a}_n)]$ on Cauchy-sequence classes; the extension is unitary with inverse $L_{-v}$. The cocycle identity gives $L_v{L}_w{\delta }_x=m_{\lambda }(w,x)m_{\lambda }(v,w+x){\delta }_{v+w+x}=m_{\lambda }(v,w)m_{\lambda }(v+w,x){\delta }_{v+w+x}=m_{\lambda }(v,w)L_{v+w}{\delta }_x\mathrm{.}$ Hence $L_v{L}_w=m_{\lambda }(v,w)L_{v+w}$ on the dense basis span and then everywhere. Also $L_0=I$ and $L_v^{\ast }=L_{-v}$, so ${\delta }_v\mapsto L_v$ extends linearly to a unital $\ast$-representation of the twisted finite-support algebra.

By [F1], this unital $\ast$-representation is contractive for the universal seminorm and extends to a representation $\mathrm{\Pi }$ of $C_{\mathrm{W}}^{\ast }(V,\omega ,\lambda )$ with $\mathrm{\Pi }({\delta }_v)=L_v$.

Fix any nonzero $v\in V$. For every real $t\mathrm{\ne }0$, $L_{tv}{\delta }_0={\delta }_{tv}$, while $L_0{\delta }_0={\delta }_0$. Since $tv\mathrm{\ne }0$, these are distinct orthonormal basis vectors, and therefore $\mathrm{\parallel }{L}_{tv}{\delta }_0-L_0{\delta }_0{\mathrm{\parallel }}_2=\sqrt{2}$ for every $t\mathrm{\ne }0$. The orbit map $t\mapsto \mathrm{\Pi }({\delta }_{tv})$ is not strongly continuous at zero. This proves that $\mathrm{\Pi }$ is nonregular although it is a representation of the universal Weyl $C^{\ast }$-algebra.