ALGEBRAIC SEMIGROUP ACTIONS I. C*-ALGEBRAS AND GROUPOIDS

The paper proves pure infiniteness under minimality by handling all basic opens of the boundary tight spectrum ∂Ê̂ in two cases and using minimality in a crucial index argument, via Lemmas 4.24 and 4.25, to assemble paradoxical bisections for arbitrary compact-open sets (Theorem 4.21) . The candidate solution only treats sets of the special form D(g,s) attached to Hs=σs(G), asserts (incorrectly) that these form a compact-open basis of ∂Ê̂, and then tries to pass to general compact-opens by covering them with such D(g,s). In the paper, basic opens are of the more general form ∂Ê̂(gC; {hiDi}) with C ranging over all constructible subgroups C (not just Hs), and the proof uses minimality to move between these constructs and to ensure indices/finiteness needed for paradoxical decompositions . The model’s omission of minimality and its unsupported basis claim break the argument for general compact-open sets.