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Some remarks on the \(C^*\)-algebras associated with subshifts. (English) Zbl 1067.46045
Summary: 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 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.

MSC:
46L05 General theory of \(C^*\)-algebras
37B10 Symbolic dynamics
46L80 \(K\)-theory and operator algebras (including cyclic theory)
47L35 Nest algebras, CSL algebras
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