  In this letter we study the eccentricity evolution of a massive black hole
(MBH) binary (MBHB) embedded in a rotating stellar cusp. Following the
observation that stars on counter-rotating (with respect to the MBHB) orbits
extract angular momentum from the binary more efficiently then their
co-rotating counterparts, the eccentricity evolution of the MBHB must depend on
the degree of co-rotation (counter-rotation) of the surrounding stellar
distribution. Using an hybrid scheme that couples numerical three-body
scatterings to an analytical formalism for the cusp-binary interaction, we
verify this hypothesis by evolving the MBHB in spherically symmetric cusps with
different fractions F of co-rotating stars. Consistently with previous works,
binaries in isotropic cusps (F=0.5) tend to increase their eccentricity, and
when F approaches zero (counter-rotating cusps) the eccentricity rapidly
increases to almost unity. Conversely, binaries in cusps with a significant
degree of co-rotation (F>0.7) tend to become less and less eccentric,
circularising quite quickly for F approaching unity. Direct N-body integrations
performed to test the theory, corroborate the results of the hybrid scheme, at
least at a qualitative level. We discuss quantitative differences, ascribing
their origin to the oversimplified nature of the hybrid approach.
