Similarity Classification and Properties of Some Extended Holomorphic Curves

For ({Omegasubseteq 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:Omegarightarrow 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.