{"title":"无界kk理论中的同伦等价","authors":"Koen van den Dungen, B. Mesland","doi":"10.2140/AKT.2020.5.501","DOIUrl":null,"url":null,"abstract":"We propose a new notion of unbounded $K\\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\\sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $\\overline{U\\!K\\!K}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $\\overline{U\\!K\\!K}(A,B)$ is isomorphic to Kasparov's $K\\!K$-theory group $K\\!K(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy relation on unbounded cycles coincides with the relation generated by operator-homotopies and addition of degenerate cycles.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"56 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Homotopy equivalence in unbounded\\nKK-theory\",\"authors\":\"Koen van den Dungen, B. Mesland\",\"doi\":\"10.2140/AKT.2020.5.501\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We propose a new notion of unbounded $K\\\\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\\\\sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $\\\\overline{U\\\\!K\\\\!K}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $\\\\overline{U\\\\!K\\\\!K}(A,B)$ is isomorphic to Kasparov's $K\\\\!K$-theory group $K\\\\!K(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy relation on unbounded cycles coincides with the relation generated by operator-homotopies and addition of degenerate cycles.\",\"PeriodicalId\":309711,\"journal\":{\"name\":\"arXiv: K-Theory and Homology\",\"volume\":\"56 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-07-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: K-Theory and Homology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/AKT.2020.5.501\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: K-Theory and Homology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/AKT.2020.5.501","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We propose a new notion of unbounded $K\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $\overline{U\!K\!K}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $\overline{U\!K\!K}(A,B)$ is isomorphic to Kasparov's $K\!K$-theory group $K\!K(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy relation on unbounded cycles coincides with the relation generated by operator-homotopies and addition of degenerate cycles.