Explicit representation-theoretic norm for the essential crossed product in the non-quasi-countable case

Develop an explicit norm for the essential crossed product of a unital inverse semigroup action on a C*-algebra, expressed in terms of the covariant representations \(\rho_\pi\times\sigma_\pi\) associated to irreducible representations \(\pi\in\widehat A\), when the inverse semigroup is not quasi-countable.

Background

The paper defines $Ess(A,S,\alpha)$ using a seminorm obtained from the family of covariant representations $\{\rho_\pi\times\sigma_\pi:\pi\in\widehat A\}$. When $S$ is quasi-countable, Theorem “reconcile ess” proves that this construction agrees with the essential crossed product $A\rtimes_{\mathrm{ess}}S$.

For non-quasi-countable inverse semigroups, the paper characterizes the ideal $N_{EL}$ through the compressed operators $$P_{[1,\pi]}(\rho_\pi\times\sigma_\pi(a^*a))P_{[1,\pi]}$$, but does not obtain an explicit norm for $A\rtimes_{\mathrm{ess}}S$ directly from the representations $\rho_\pi\times\sigma_\pi$.