This question is motivated by a similar question recently posed at MO: https://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras
In this setting, let $B$ be a Banach algebra generated by projections, meaning that the span of all idempotents in $B$ is norm dense. Is the second dual of $B$ endowed with either Arens product generated by projections?