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

Let 𝒜 {\mathcal{A}} be a complex unital Banach algebra and let R ⊆ 𝒜 {R\subseteq\mathcal{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 {R^{D}} is constructed as an extension of R to axiomatically study the accumulation of σ R ⁢ ( a ) {\sigma_{R}(a)} for any a ∈ 𝒜 {a\in\mathcal{A}} . At last, several illustrative examples on Banach algebra and operator algebra are provided.