Connection Between Cohn Path Algebra and C*-Algebra Through Leavitt Path Algebra Nugroho Dwi Widodo, Shely Mutiara Maghfira, Rizky Rosjanuardi, Sumanang Muhtar Gozali
Universitas Pendidikan Indonesia
Department of Mathematics Education FPMIPA
Abstract
For any directed graph \(E\) and any field \(K\) we can produce Leavitt path algebra from Cohn path algebra \(C_{K}(E)\). The result of investigation by Abrams, Pere Ara, dan Molina, we could choose a directed graph \(F\) such that Cohn path algebra and Leavitt path algebra are isomorphic which is \(C_{K}(E)\)\(\cong\)\(L_{K}(F)\). As field \(K\)=\(\bf C\) (complex number), according Tomforde^s discussion, that Leavitt path algebra is isomorphic to a dense *-subalgebra, in particular \(L_{\bf C}(F)\)\(\cong\)\(C^{*}(F)\). Based on both connection, is there any connection between Cohn path algebra and \(C^{*}\)-algebra? Through each connection between Cohn path algebra and \(C^{*}\)-algebra with Leavitt path algebra, we obtained \(C_{\bf C}(E)\)\(\cong\)\(C^{*}(F)\), so that Cohn path algebra could be viewed as graph algebra from \(C^{*}(F)\), which is \(C^{*}\)-algebra for a directed graph \(F\), with graph \(F\) was graph made from directed graph \(E\) by adding some edges and vertex based on certain conditions.
