{"title":"有限型von Neumann代数中的非交换泛函演算","authors":"Piotr Niemiec","doi":"10.3318/pria.2016.116.10","DOIUrl":null,"url":null,"abstract":"A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f_1,f_2,... of functions that converge pointwise to 0 transform into sequences f_1[T],f_2[T],... of operators in M that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T which acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all function from the aforementioned class. Some conclusions concerning so-called operator-spectra of such operators are drawn and a new variation of the spectral theorem for them is formulated.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"169 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2013-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Non-commutative functional calculus in finite type I von Neumann algebras\",\"authors\":\"Piotr Niemiec\",\"doi\":\"10.3318/pria.2016.116.10\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f_1,f_2,... of functions that converge pointwise to 0 transform into sequences f_1[T],f_2[T],... of operators in M that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T which acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all function from the aforementioned class. Some conclusions concerning so-called operator-spectra of such operators are drawn and a new variation of the spectral theorem for them is formulated.\",\"PeriodicalId\":434988,\"journal\":{\"name\":\"Mathematical Proceedings of the Royal Irish Academy\",\"volume\":\"169 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2013-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Proceedings of the Royal Irish Academy\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3318/pria.2016.116.10\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Proceedings of the Royal Irish Academy","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3318/pria.2016.116.10","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Non-commutative functional calculus in finite type I von Neumann algebras
A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f_1,f_2,... of functions that converge pointwise to 0 transform into sequences f_1[T],f_2[T],... of operators in M that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T which acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all function from the aforementioned class. Some conclusions concerning so-called operator-spectra of such operators are drawn and a new variation of the spectral theorem for them is formulated.