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<p>In this paper, by employing the quantitative <i>K</i>-theory, we introduce the quantitative coarse Baum-Connes conjecture with coefficients (or QCBC, for short) for proper metric spaces, which provides an algorithm to compute the quantitative indices of elliptic differential operators on complete Riemannian manifolds. On the one hand, QCBC refines the coarse Baum-Connes conjecture with coefficients. On the other hand, we prove that QCBC can be derived from the coarse Baum-Connes conjecture with different coefficients, which provides many examples satisfying QCBC. Besides, we show that QCBC can be reduced to the case of sparse spaces.</p>

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On the quantitative coarse Baum-Connes conjecture with coefficients

  • 期刊论文

  • 发表日期：

    2026年1月26日

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  • Jianguo Zhang

摘要

In this paper, by employing the quantitative K-theory, we introduce the quantitative coarse Baum-Connes conjecture with coefficients (or QCBC, for short) for proper metric spaces, which provides an algorithm to compute the quantitative indices of elliptic differential operators on complete Riemannian manifolds. On the one hand, QCBC refines the coarse Baum-Connes conjecture with coefficients. On the other hand, we prove that QCBC can be derived from the coarse Baum-Connes conjecture with different coefficients, which provides many examples satisfying QCBC. Besides, we show that QCBC can be reduced to the case of sparse spaces.