  We introduce and systematically study linear sofic groups and linear sofic
algebras. This generalizes amenable and LEF groups and algebras. We prove that
a group is linear sofic if and only if its group algebra is linear sofic. We
show that linear soficity for groups is a priori weaker than soficity but
stronger than weak soficity. We also provide an alternative proof of a result
of Elek and Szabo which states that sofic groups satisfy Kaplansky's direct
finiteness conjecture.
