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SUMMARY:Equivalence of categories of KK-theory or E-theory for C*-algebras
  over topological spaces by reflection functors - Nanami Hashimoto (Keio U
 niversity)
DTSTART:20250716T155000Z
DTEND:20250716T161000Z
UID:TALK233260@talks.cam.ac.uk
DESCRIPTION:In this talk\, for finite $T_0$-spaces $X$ and $Y$ satisfying 
 certain conditions\, I will introduce reflection functors between the cate
 gories $\\mathfrak{C^*sep}(X)$ and $\\mathfrak{C^*sep}(Y)$ of separable C*
 -algebras over $X$ and $Y$. By the universal properties of KK-theory and E
 -theory\, these functors induce reflection functors between the categories
  $\\mathfrak{KK}(X)$ and $\\mathfrak{KK}(Y)$ of Kirchberg's ideal-related 
 KK-theory\, as well as between the categories $\\mathfrak{E}(X)$ and $\\ma
 thfrak{E}(Y)$ of Dadarlat and Meyer's ideal-related E-theory. I will show 
 that the reflection functors between $\\mathfrak{KK}(X)$ and $\\mathfrak{K
 K}(Y)$ define an equivalence between the bootstrap categories $\\mathcal{B
 }(X)$ and $\\mathcal{B}(Y)$. I will also show that the categories $\\mathf
 rak{E}(X)$ and $\\mathfrak{E}(Y)$\, and the E-theoretic bootstrap categori
 es $\\mathcal{B}_{\\mathrm{E}}(X)$ and $\\mathcal{B}_{\\mathrm{E}}(Y)$ are
  equivalent via the reflection functors. The reflection functors introduce
 d here are reminiscent of the BGP-reflection functors in the representatio
 n theory of quivers. As a consequence\, I will show that $\\mathcal{B}(X)$
  and $\\mathcal{B}(Y)$\, $\\mathfrak{E}(X)$ and $\\mathfrak{E}(Y)$\, and $
 \\mathcal{B}_{\\mathrm{E}}(X)$ and $\\mathcal{B}_{\\mathrm{E}}(Y)$ are res
 pectively equivalent whenever the undirected graphs associated with $X$ an
 d $Y$ are the same tree. If time permits\, I will also discuss how reflect
 ion functors can be applied to construct K-theoretic invariants that satis
 fy the Universal Coefficient Theorems for ideal-related KK-theory and E-th
 eory.
LOCATION:External
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