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In the first half of this chapter, we will introduce the notion of a relative version of Morita equivalence of \(C^*\) -algebras to study pairs \((\mathcal {O}_A, \mathcal {D}_A)\) and \((\mathcal {O}_A\otimes \mathcal {K}(H), \mathcal {D}_A\otimes \mathcal {C}(H))\) of Cuntz–Krieger algebras with its canonical Cartan subalgebras and its stabilizations, where \(\mathcal {K}(H)\) denotes the \(C^*\) -algebra of compact operators on a separable infinite-dimensional Hilbert space \(H(=\ell ^2(\mathbb {N}))\) and \(\mathcal {C}(H)\) its maximal commutative \(C^*\) -subalgebra of \(\mathcal {K}(H)\) consisting of diagonal operators on H.

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Strong Shift Equivalence, Flow Equivalence and Cuntz–Krieger Algebras

  • 章节

  • 发表日期：

    2025年

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  • Kengo Matsumoto

摘要

In the first half of this chapter, we will introduce the notion of a relative version of Morita equivalence of \(C^*\) -algebras to study pairs \((\mathcal {O}_A, \mathcal {D}_A)\) and \((\mathcal {O}_A\otimes \mathcal {K}(H), \mathcal {D}_A\otimes \mathcal {C}(H))\) of Cuntz–Krieger algebras with its canonical Cartan subalgebras and its stabilizations, where \(\mathcal {K}(H)\) denotes the \(C^*\) -algebra of compact operators on a separable infinite-dimensional Hilbert space \(H(=\ell ^2(\mathbb {N}))\) and \(\mathcal {C}(H)\) its maximal commutative \(C^*\) -subalgebra of \(\mathcal {K}(H)\) consisting of diagonal operators on H.