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<p>Ghostly ideals are among the most mysterious objects in coarse index theory. In this paper we show that if a metric space <i>X</i> with bounded geometry admits a coarse embedding into an <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\ell ^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-space (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(1 \le p &lt; \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>&lt;</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>), then the canonical inclusion from any geometric ideal to the corresponding ghostly ideal induces an isomorphism in <i>K</i>-theory. As consequences, we deduce that such spaces satisfy the relative coarse Baum-Connes conjectures introduced in [<CitationRef CitationID="CR12">12</CitationRef>], as well as the operator norm localization property for finite rank projections (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(ONL_{\mathcal {P}_{Fin}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mi>N</mi> <msub> <mi>L</mi> <msub> <mi mathvariant="script">P</mi> <mrow> <mi mathvariant="italic">Fin</mi> </mrow> </msub> </msub> </mrow> </math></EquationSource> </InlineEquation>) as introduced in [<CitationRef CitationID="CR1">1</CitationRef>].</p>

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K-theory of ghostly ideals for \(\ell ^p\)-coarsely embeddable spaces

  • 期刊论文

  • 发表日期：

    2026年6月17日

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  • Liang Guo,
  • Kang Li,
  • Qin Wang

摘要

Ghostly ideals are among the most mysterious objects in coarse index theory. In this paper we show that if a metric space X with bounded geometry admits a coarse embedding into an \(\ell ^p\) ℓ p -space ( \(1 \le p < \infty \) 1 ≤ p < ∞ ), then the canonical inclusion from any geometric ideal to the corresponding ghostly ideal induces an isomorphism in K-theory. As consequences, we deduce that such spaces satisfy the relative coarse Baum-Connes conjectures introduced in [12], as well as the operator norm localization property for finite rank projections ( \(ONL_{\mathcal {P}_{Fin}}\) O N L P Fin ) as introduced in [1].