  In this article, we present some results and constructions about the
Chiu-Tamarkin invariant motivated by the idempotence of microlocal kernels,
including: (1) a natural explanation for the definition of the
$\mathbb{Z}/\ell$-equivariant Chiu-Tamarkin invariant; (2) a graded commutative
product on the non-equivariant Chiu-Tamarkin invariant; and (3) a construction
of the $S^1$-equivariant Chiu-Tamarkin invariant. As applications, we: (1)
construct a sequence of symplectic capacities $(\overline{c}_k)_{k\in
\mathbb{N}}$ and prove that it coincides with the symplectic capacities
$({c}_k)_{k\in \mathbb{N}}$ we defined using the $\mathbb{Z}/\ell$-equivariant
Chiu-Tamarkin invariant under certain conditions; and (2) prove a Viterbo
isomorphism. In the Appendix, we provide a proof of admissibility for all open
sets in a cotangent bundle under the setup of triangulated categories.
