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Semigroups S for which the Banach algebra is injective are investigated and an application to the work of O. Yu. Aristov is described.
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].
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