Mishiba's Conjecture on the Coaction of $\mathrm{\infty }$-adic Multiple Zeta Values

We study multiple zeta values in positive characteristic. We construct, using $\mathrm{\infty }$-adic multiple zeta values, a coaction of the $\mathrm{\infty }$-adic multiple zeta value algebra modulo ${\zeta }_A(q-1)$ on the $\mathrm{\infty }$-adic multiple zeta value algebra itself, and prove that it agrees with Mishiba’s coaction constructed via special values of Carlitz multiple polylogarithms. We also determine the coaction on the $\star$-inverse values and show that the antipode on the $\mathrm{\infty }$-adic multiple zeta value algebra modulo ${\zeta }_A(q-1)$ is given by the $\star$-inverse operation. In particular, these results prove Mishiba’s conjecture concerning the coaction and the antipode for $\mathrm{\infty }$-adic multiple zeta values.