Extendable Shift Maps and the Canonical Map on Generalized Countable Markov Shifts
We obtain an operator-algebraic characterization of when the shift map can be continuously extended from a standard countable Markov shift $Σ_A$ to its generalized countable Markov shift $X_A$, a compactification of $Σ_A$. We show that a canonical unital completely positive map on the Exel--Laca algebra $O_A$ can be defined exactly when the shift extends continuously from $Σ_A$ to $X_A$, and that in this case its restriction to the maximal commutative C$^*$-subalgebra $D_A$ is a $*$-endomorphism to which the extended shift is dual. We also give a criterion on the matrix $A$ itself, in terms of the convergence of its columns, and exhibit classes of transition matrices satisfying these equivalent conditions. Column-finiteness of $A$ is necessary for the existence of the continuous extension. Finally, we construct an example in which the set of empty words is a Cantor set and the extended dynamics on it is the inverse of the dyadic odometer.