Uniqueness Theorems and Ideal Structure for Leavitt Path Algebras

We prove Leavitt path algebra versions of the two uniqueness theorems of graph C*-algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their simplicity. We also use these results to give a proof of the fact that for any graph E the Leavitt path algebra $L_\mathbb{C}(E)$ embeds as a dense *-subalgebra of the graph C*-algebra C*(E). This embedding has consequences for graph C*-algebras, and we discuss how we obtain new information concerning the construction of C*(E).Comment: 34 pages, uses XY-pic. New version comments: Some small typos corrected. This is the final version to appear in the Journal of Algebr

Vector spaces with an order unit

We develop a theory of ordered *-vector spaces with an order unit. We prove fundamental results concerning positive linear functionals and states, and we show that the order (semi)norm on the space of self-adjoint elements admits multiple extensions to an order (semi)norm on the entire space. We single out three of these (semi)norms for further study and discuss their significance for operator algebras and operator systems. In addition, we introduce a functorial method for taking an ordered space with an order unit and forming an Archimedean ordered space. We then use this process to describe an appropriate notion of quotients in the category of Archimedean ordered spaces.Comment: 38 pages, uses XY-pic, Version 2 comments: minor typos corrected.; Version 3 Comments: minor typos corrected; Version 4 Comments: minor typos corrected, hypothesis of Archimedean added to Theorem 4.22, To appear in Indiana Univ. Math.