Extendable Shift Maps and Weighted Endomorphisms on Generalized Countable Markov Shifts

We obtain an operator algebraic characterization for when we can continuously extend the shift map from a standard countable Markov shift ${\mathrm{\Sigma }}_A$ to its respective generalized countable Markov shift $X_A$ (a compactification of ${\mathrm{\Sigma }}_A$). When the shift map is continuously extendable, we obtain explicit formulas for the spectral radius of weighted endomorphisms $a\alpha$, where $\alpha$ is dual to the shift map and conjugated to $\mathrm{\Theta }(f)=f\circ \sigma$ on $C(X_A)$, extending a theorem of Kwaśniewski and Lebedev from finite to countable alphabets.