  Given a Banach space X and a bounded linear operator T on X, a subspace Y of
X is almost invariant under T if TY is a subspace of Y+F for some
finite-dimensional ``error'' F. In this paper, we study subspaces that are
almost invariant under every operator in an algebra A of operators acting on X.
We show that if A is norm closed then the dimensions of ``errors''
corresponding to operators in A must be uniformly bounded. Also, if A is
generated by a finite number of commuting operators and has an almost invariant
half-space (that is, a subspace with both infinite dimension and infinite
codimension) then A has an invariant half-space.
