  Motivated by a result of Ky Fan in 1965, we establish a characterization of a
left amenable F-algebra (which includes the group algebra and the Fourier
algebra of a locally compact group and quantum group algebras, or more
generally the predual algebra of a Hopf von Neumann algebra) in terms of a
finite dimensional invariant subspace property. This is done by first revealing
a fixed point property for the semigroup of norm one positive linear func-
tionals in the algebra. Our result answers an open question posted in Tokyo in
1993 by the first author (see [25, Problem 5]). We also show that the left
amenability of an ideal in an F-algebra may determine the left amenability of
the algebra.
