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M/ D S m ˝ Id S . Obviously, ˚ satisfies the required conditions. u Remark 1. We keep the notation of the previous proposition. A straightforward computation shows that F is the positive definite function F˚ constructed from ˚. Proposition 3. Let F be a Borel positive definite function on G such that FjX D 1. fjF j > "g/ < C1. Let ˚ be the completely positive map defined by F . hM; eA i/ generated by the finite projections of hM; eA i. Proof. F / by F . jT j/ of jT j relative to t; C1Œ is finite.

Math. 525, 17–53 (2000). 065. 065 14. : C -algebras over topological spaces: filtrated K-theory. Canad. J. Math. 64(2), 368–408 (2012). 4153/CJM-2011-061-x. 4153/CJM2011-061-x 15. : Strongly self-absorbing C -algebras. Trans. Amer. Math. Soc. 359(8), 3999–4029 (2007). 1090/S0002-9947-07-04173-6. 1090/ S0002-9947-07-04173-6 Chapter 3 Projective Dimension in Filtrated K-Theory Rasmus Bentmann Abstract Under mild assumptions, we characterise modules with projective resolutions of length n 2 N in the target category of filtrated K-theory over a finite topological space in terms of two conditions involving certain Tor-groups.

Qn0 / 1 : 26 C. Qn / is an increasing sequence of Borel symmetric and bounded subsets of Q with [n Qn D Q. Let Gn be the subgroupoid of G generated by Qn , that is Gn D [k 0 Qnk , where we put Qn0 D X . We observe that Qn is a treeing for Gn . Denote by n the associated conditionally negative definite function on Gn . Since Qn 1 Qn , we have . n /jGn 1 Ä n 1: Given two integers k and N , we set Ak;N D f 2 Gk W k. Ak;k /k is an increasing sequence of Borel subsets of G with [k Ak;k D G. We fix k.

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Encyclopaedie der mathematischen Wissenschaften und Anwendungen. Algebra und Zahlentheorie

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