The one-dimensional trivial VOA and its Zhu algebra

The only nonzero mode of $Y(\mathbf{1},z)={\mathrm{id}}_V$ is ${\mathbf{1}}_{(-1)}={\mathrm{id}}_V$. Truncation holds; creation gives $Y(\mathbf{1},z)\mathbf{1}=\mathbf{1}=e^{zT}\mathbf{1}$; translation covariance has both sides zero; and locality holds because the only fields commute. All $L(m)=0$, so the Virasoro relations hold with $c=0$, $L(-1)=T=0$, and $V_0=V=\mathbb{C}\mathbf{1}$, with all other weight spaces zero. Thus the CFT-type axioms hold. The only subspaces of a one-dimensional vector space are zero and itself, so $V$ is simple.

Every $a=\lambda \mathbf{1}$ has weight zero. By [F3], $a\circ b=\lambda \mu {\mathrm{Res}}_z{z}^{-2}\mathbf{1}=0,a\ast b=\lambda \mu {\mathrm{Res}}_z{z}^{-1}\mathbf{1}=\lambda \mu \mathbf{1.}$ Hence $O(V)=0$, the quotient is $V\cong \mathbb{C}$, and its product is scalar multiplication with unit $[\mathbf{1}]$. These computations verify the quotient and product directly in this example.