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In this paper, we give a new geometric condition in terms of measured asymptotic expanders to ensure rigidity of Roe algebras. Consequently, we obtain the rigidity for all bounded geometry spaces that coarsely embed into some   $L^p$  -space for   $p\in [1,\infty )$  . Moreover, we also verify rigidity for the box spaces constructed by Arzhantseva–Tessera and Delabie–Khukhro even though they do  not  coarsely embed into any   $L^p$  -space. The key step in our proof of rigidity is showing that a block-rank-one (ghost) projection on a sparse space   $X$   belongs to the Roe algebra   $C^{\ast }(X)$   if and only if   $X$   consists of (ghostly) measured asymptotic expanders. As a by-product, we also deduce that ghostly measured asymptotic expanders are new sources of counterexamples to the coarse Baum–Connes conjecture.

Measured Asymptotic Expanders and Rigidity for Roe Algebras