Some remarks on the $C^*$-algebras associated with subshifts

We point out incorrect lemmas in some papers regarding the $C^*$-algebras associated with subshifts written by the second named author. To recover the incorrect lemmas and the affected main results, we will describe an alternative construction of $C^*$-algebras associated with subshifts. The resulting $C^*$-algebras are generally different from the originally constructed $C^*$-algebras associated with subshifts and they fit the mentioned papers including the incorrect results. The simplicity conditions and the K-theory formulae for the originally constructed $C^*$-algebras are described. We also introduce a condition called $(*)$ for subshifts such that under this condition the new $C^*$-algebras and the original $C^*$-algebras are canonically isomorphic to each other. We finally present a subshift for which the two kinds of algebras have different K-theory groups.