Theorem. the eigenspace decomposition of a module [fgap-000Z]
✍️source

    Let \(z\) be a central involution in \(A\), let \(e_+,e_-\) be its associated idempotents, and let \(M\) be a left \(A\)-module. Left multiplication defines \(A\)-linear projections \[ p_\pm :M\longrightarrow M,\qquad p_\pm (m)=e_\pm m. \] They satisfy \[ p_\pm ^2=p_\pm ,\qquad p_++p_-=\mathrm {id}_M,\qquad p_+p_-=p_-p_+=0. \] Their ranges and kernels are \[ \begin {aligned} \operatorname {range}(p_+)&=e_+M,& \ker (p_+)&=\operatorname {range}(p_-)=e_-M,\\ \operatorname {range}(p_-)&=e_-M,& \ker (p_-)&=\operatorname {range}(p_+)=e_+M. \end {aligned} \] Consequently, if \[ M_+=e_+M,\qquad M_-=e_-M, \] then \[ M=M_+\oplus M_-. \] These summands are precisely the two eigenspaces for the action of \(z\): \[ M_+=\{m\in M:zm=m\},\qquad M_-=\{m\in M:zm=-m\}. \]

    Proof. 

      The idempotent and mixed-product identities from the two idempotents of a central involution give the displayed projection identities after acting on \(M\). Centrality of \(e_\pm \) makes \(p_\pm \) \(A\)-linear. Their ranges are \(e_\pm M\) by definition. If \(p_+(m)=0\), then \[ m=(p_++p_-)(m)=p_-(m), \] so \(m\) lies in the range of \(p_-\). Conversely, \(p_+p_-=0\) shows that every element in that range lies in \(\ker (p_+)\). This proves the first kernel-range equality; the second is symmetric. The identity \(p_++p_-=\mathrm {id}_M\) now gives the displayed direct sum.

      Direct calculation gives \(ze_+=e_+\) and \(ze_-=-e_-\), proving one eigenspace inclusion in each case. Conversely, if \(zm=m\), then \[ e_+m=h(m+zm)=2hm=m. \] If \(zm=-m\), the analogous calculation gives \(e_-m=m\).

      Idempotents as projections and complementary orthogonal idempotents as direct decompositions are recorded in [webb2007finite, exercise 2.7, p. 15]; see also [sengupta2010representations, props. 4.5.1--4.5.3, pp. 63--67].