On the separability problem for isometric shifts on C(X)

We provide examples of nonseparable spaces X for which C(X) admits an isometric shift, which solves in the negative a problem proposed by Gutek et al. [A. Gutek, D. Hart, J. Jamison, M. Rajagopalan, Shift operators on Banach spaces, J. Funct. Anal. 101 (1991) 97–119].