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

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {A} be a complex unital Banach algebra and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>⊆</m:mo> <m:mi mathvariant="script">𝒜</m:mi> </m:mrow> </m:math> {R} 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, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>R</m:mi> <m:mi>D</m:mi> </m:msup> </m:math> {RD} is constructed as an extension of R to axiomatically study the accumulation of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>R</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>a</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {σR(a)} for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">𝒜</m:mi> </m:mrow> </m:math> {a} . At last, several illustrative examples on Banach algebra and operator algebra are provided.