Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum

Let A be a complex unital Banach algebra and let R A be a non-empty set. This paper defines the property such that R is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity R with (CID) property, R^D is constructed as an extension of R to axiomatically study the accumulation of σ R (a) for any a ∈ A. At last, several illustrative examples on Banach algebra and operator algebra are provided.