Assumption: Old microcausal product

The full old compact polynomial microcausal algebra ${\mathcal{P}}_{\mathrm{mc}}$ is included in ${\mathcal{P}}_{\Sigma }$. Its functional derivatives satisfy the standard microcausal wavefront condition: no nonzero joint covector is entirely future causal or entirely past causal, where zero individual covectors are allowed in those closed causal cones.

The inherited old product ${\star }_H$ is defined on ${\mathcal{P}}_{\mathrm{mc}}\otimes {\mathcal{P}}_{\mathrm{mc}}$, has classical order zero, and has its unchanged coefficients

For the reversal statement we use the inherited Hermitian old convention

No new continuity property of ${\star }_H$ is introduced.