Application on Unitaries in a Simple πͺ β -Algebra of Tracial Rank One
Journal Title: International Journal of Engineering and Science Invention - Year 2018, Vol 7, Issue 2
Abstract
Let π΄ be a unital separable simple infinite dimensional πΆ β -algebra,with tracial rank nomore than one and with the tracial state space π(π΄). Let π(π΄) be the unitary group ofπ΄. Suppose that π’ 2 β π0 π΄ , whenπ0(π΄) bethe connected component of π(π΄) containing the identity. Weshow that, for any π > 0, there exists a selfadjoint element π 2 β π΄π .π 2 such that π’ 2 βexp(ππ 2 ) < π. We also show the problem when π’ 2 can be approximated by unitaries in π΄with finite spectrum. Denote by πΆπ(π΄) the closure of the subgroup of unitary group of π(π΄) generated byits commutators. It is known that πΆπ(π΄) β π0(π΄). Denote by π 2 the affine function onπ(π΄) defined by π 2(π) = π(π 2 ). We show that π’ 2 can be approximated by unitaries in π΄withfinite spectrum if and only if π’ 2 β πΆπ(π΄) and π’ 2π + π’ 2π β , π π’ 2π β π’ 2π β β ππ΄ πΎ0 π΄ for allπ β₯ 1. Examples are given that there are unitaries in πΆπ π΄ which can not be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence ofamenability.Let π΄ be a unital separable simple infinite dimensional πΆ β -algebra,with tracial rank nomore than one and with the tracial state space π(π΄). Let π(π΄) be the unitary group ofπ΄. Suppose that π’ 2 β π0 π΄ , whenπ0(π΄) bethe connected component of π(π΄) containing the identity. Weshow that, for any π > 0, there exists a selfadjoint element π 2 β π΄π .π 2 such that π’ 2 βexp(ππ 2 ) < π. We also show the problem when π’ 2 can be approximated by unitaries in π΄with finite spectrum. Denote by πΆπ(π΄) the closure of the subgroup of unitary group of π(π΄) generated byits commutators. It is known that πΆπ(π΄) β π0(π΄). Denote by π 2 the affine function onπ(π΄) defined by π 2(π) = π(π 2 ). We show that π’ 2 can be approximated by unitaries in π΄withfinite spectrum if and only if π’ 2 β πΆπ(π΄) and π’ 2π + π’ 2π β , π π’ 2π β π’ 2π β β ππ΄ πΎ0 π΄ for allπ β₯ 1. Examples are given that there are unitaries in πΆπ π΄ which can not be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence ofamenability.
Authors and Affiliations
Abd Ellteaf YahiA, Andbelgiss Abd ElaziZ, Hanana Elnage, Shreefa Osman
Keywords
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