Theorem. the central-idempotent algebra product [fgap-0010]
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    For a central involution \(z\) and its associated idempotents, put \[ A_+=e_+A,\qquad A_-=e_-A. \] Because \(e_+\) and \(e_-\) are central, these are two-sided ideals and \(e_\pm A=Ae_\pm \). Each is a unital \(R\)-algebra in its own right, with unit \(e_\pm \) and structural map \(r\mapsto re_\pm \). There is an \(R\)-algebra isomorphism \[ \begin {aligned} \Phi :A&\longrightarrow A_+\times A_-, &a&\longmapsto (e_+a,e_-a),\\ \Psi :A_+\times A_-&\longrightarrow A, &(x,y)&\longmapsto x+y. \end {aligned} \]

    Proof. 

      For \(a,b\in A\), \(a(e_\pm b)=e_\pm (ab)\) and \((e_\pm b)a=e_\pm (ba)\), so \(A_\pm \) are two-sided ideals. If \(x=e_+a\in A_+\), then \(e_+x=x\); thus \(e_+\) is the unit of \(A_+\), and similarly for \(A_-\).

      Mixed products vanish: \[ (e_+a)(e_-b)=e_+e_-ab=0, \] and likewise in the reverse order. It follows that \(\Phi \) and \(\Psi \) preserve multiplication. Finally, \[ \Psi \Phi (a)=(e_++e_-)a=a, \] while \(\Phi \Psi (x,y)=(x,y)\) because the matching idempotent fixes each factor and the other annihilates it.

      The use of local units on idempotent-generated ideals follows the standard discussion in [sengupta2010representations, secs. 4.5--4.6, pp. 63--73].