Direct CP, Lepton Flavor, and Isospin Asymmetries in the Decays B -> K((*))l(+)l(-)

We measure rate asymmetries for the rare decays B {yields} K{sup (*)}{ell}{sup +}{ell}{sup -}, where {ell}{sup +}{ell}{sup -} is either e{sup +}e{sup -} or {mu}{sup +}{mu}{sup -}, using a sample of 384 million B{bar B} events collected with the BABAR detector at the PEP-II e{sup +}e{sup -} collider. We find no evidence for direct CP or lepton-flavor asymmetries. For dilepton masses below the J/{psi} resonance, we find evidence for unexpectedly large isospin asymmetries in both B {yields} K{ell}{sup +}{ell}{sup -} and B {yields} K*{ell}{sup +}{ell}{sup -} which differ respectively by 3.2{sigma} and 2.7{sigma}, including systematic uncertainties, from the Standard Model expectations.