Free Wick algebra and equation ideal

Use the polynomial microcausal Wick presentation of the free scalar algebra. Choose a Hadamard bisolution ${\Delta }_{+}$ once, with

For the polynomial entries used here, the Wick product is

The sum is finite for each polynomial pair. This is the standard free presentation FR (Sections 4–5); no closure assertion for an arbitrary class of nonpolynomial smooth functionals is needed.

Let ${\mathcal{A}}_{\mathrm{off}}$ denote this off-shell polynomial algebra. Since ${\Delta }_{+}$ is a bisolution,

The pointwise ideal generated by all $F_{P_{m}u}$, $u\in \mathcal{D}(M)$, is therefore also a star ideal, denoted ${\mathfrak{J}}_0$. Put

Elements of ${\mathfrak{J}}_0$ vanish on smooth free solutions. In particular,

The linear output topology is the quotient test topology on $\mathcal{D}(M)/P_m\mathcal{D}(M)$.