  In this study, we determined the effect of doping with the noble metal Ag on
Ir$_2$Te$_4$ superconductors. Based on the resistivity, magnetization, and heat
capacity, we explored the changes in the superconductivity and charge density
wave for Cu$_{1-x}$Ag$_x$Ir$_2$Te$_4$ as a function of isoelectric
substitution. We assessed a complete set of competing states from suppressed
charge density wave in the low doping region to superconductor in the middle
doping region and re-entrant charge density wave in the high doping region,
thereby obtaining an electronic phase diagram, where the superconducting dome
was near bipartite charge density wave regions with a maximum superconducting
temperature Tc of about 2.93 K at an Ag doping level of 12%. The lower H$_{c1}$
and upper H$_{c2}$ critical magnetic fields were determined for some
representative samples in the Cu$_{1-x}$Ag$_x$Ir$_2$Te$_4$ series based on
magnetization and resistivity measurements, respectively. We showed that
H$_{c1}$ decreased whereas H$_{c2}$ increased as the doping content increased.
The specific heat anomalies at the superconducting transitions $\Delta
C_{el}/\gamma T_c$ for representative samples comprising
Cu$_{0.92}$Ag$_{0.08}$Ir$_2$Te$_4$, Cu$_{0.88}$Ag$_{0.12}$Ir$_2$Te$_4$, and
Cu$_{0.85}$Ag$_{0.15}$Ir$_2$Te$_4$ were approximately 1.40, 1.44, and 1.42,
respectively, which are all near the Bardeen-Cooper-Schrieffer value of 1.43
and they indicate bulk superconductivity in these compounds.
