Similarity Classification and Properties of Some Extended Holomorphic Curves

Abstract

For ${\Omega\subseteq \mathbb{C}}$ a connected open set, and ${{\mathcal U}}$ a unital Banach algebra (or a unital C*-algebra), let ${{\xi (U)}}$ and ${ P({\mathcal U})}$ denote the sets of all idempotents and projections in ${{\mathcal U}}$, respectively. If ${e:\Omega\rightarrow \xi ({\mathcal U})}$ (resp.${P({\mathcal U}))}$ is a holomorphic ${{\mathcal U}}$-valued map, then e is called an extended holomorphic curve on ${ \xi ({\mathcal U})}$ (resp. ${P({\mathcal U})}$). In this article, we focus on discussing the similarity classification problem of extended holomorphic curves. First, we introduce the definition of the commutant of extended holomorphic curves. By using K ₀-group of the commutant of the extended holomorphic curve, we characterize the curve which has unique finite (SI) decomposition up to similarity. Subsequently, we also obtain a similarity classification theorem. Second, we also discuss the unitary equivalence problem of some curves with respect to inductive limit C*-algebras.