Application on Unitaries in a Simple π‘ͺ βˆ— -Algebra of Tracial Rank One

Journal Title: International Journal of Engineering and Science Invention - Year 2018, Vol 7, Issue 2

Abstract

Let 𝐴 be a unital separable simple infinite dimensional 𝐢 βˆ— -algebra,with tracial rank nomore than one and with the tracial state space 𝑇(𝐴). Let π‘ˆ(𝐴) be the unitary group of𝐴. Suppose that 𝑒 2 ∈ π‘ˆ0 𝐴 , whenπ‘ˆ0(𝐴) bethe connected component of π‘ˆ(𝐴) containing the identity. Weshow that, for any πœ– > 0, there exists a selfadjoint element 𝑕 2 ∈ 𝐴𝑠.π‘Ž 2 such that 𝑒 2 βˆ’exp(𝑖𝑕 2 ) < πœ–. We also show the problem when 𝑒 2 can be approximated by unitaries in 𝐴with finite spectrum. Denote by πΆπ‘ˆ(𝐴) the closure of the subgroup of unitary group of π‘ˆ(𝐴) generated byits commutators. It is known that πΆπ‘ˆ(𝐴) βŠ‚ π‘ˆ0(𝐴). Denote by π‘Ž 2 the affine function on𝑇(𝐴) defined by π‘Ž 2(𝜏) = 𝜏(π‘Ž 2 ). We show that 𝑒 2 can be approximated by unitaries in 𝐴withfinite spectrum if and only if 𝑒 2 ∈ πΆπ‘ˆ(𝐴) and 𝑒 2𝑛 + 𝑒 2𝑛 βˆ— , 𝑖 𝑒 2𝑛 βˆ’ 𝑒 2𝑛 βˆ— ∈ 𝜌𝐴 𝐾0 𝐴 for all𝑛 β‰₯ 1. Examples are given that there are unitaries in πΆπ‘ˆ 𝐴 which can not be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence ofamenability.Let 𝐴 be a unital separable simple infinite dimensional 𝐢 βˆ— -algebra,with tracial rank nomore than one and with the tracial state space 𝑇(𝐴). Let π‘ˆ(𝐴) be the unitary group of𝐴. Suppose that 𝑒 2 ∈ π‘ˆ0 𝐴 , whenπ‘ˆ0(𝐴) bethe connected component of π‘ˆ(𝐴) containing the identity. Weshow that, for any πœ– > 0, there exists a selfadjoint element 𝑕 2 ∈ 𝐴𝑠.π‘Ž 2 such that 𝑒 2 βˆ’exp(𝑖𝑕 2 ) < πœ–. We also show the problem when 𝑒 2 can be approximated by unitaries in 𝐴with finite spectrum. Denote by πΆπ‘ˆ(𝐴) the closure of the subgroup of unitary group of π‘ˆ(𝐴) generated byits commutators. It is known that πΆπ‘ˆ(𝐴) βŠ‚ π‘ˆ0(𝐴). Denote by π‘Ž 2 the affine function on𝑇(𝐴) defined by π‘Ž 2(𝜏) = 𝜏(π‘Ž 2 ). We show that 𝑒 2 can be approximated by unitaries in 𝐴withfinite spectrum if and only if 𝑒 2 ∈ πΆπ‘ˆ(𝐴) and 𝑒 2𝑛 + 𝑒 2𝑛 βˆ— , 𝑖 𝑒 2𝑛 βˆ’ 𝑒 2𝑛 βˆ— ∈ 𝜌𝐴 𝐾0 𝐴 for all𝑛 β‰₯ 1. Examples are given that there are unitaries in πΆπ‘ˆ 𝐴 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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  • EP ID EP396408
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How To Cite

Abd Ellteaf YahiA, Andbelgiss Abd ElaziZ, Hanana Elnage, Shreefa Osman (2018). Application on Unitaries in a Simple π‘ͺ βˆ— -Algebra of Tracial Rank One. International Journal of Engineering and Science Invention, 7(2), 51-59. https://europub.co.uk/articles/-A-396408