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A bstract We present measurements of the branching fractions for the decays B → Kμ + μ − and B → Ke + e − , and their ratio ( R K ), using a data sample of 711 fb − 1 that contains 772 × 10 6 $$ B\overline{B} $$ B B ¯ events. The data were collected at the ϒ(4 S ) resonance with the Belle detector at the KEKB asymmetric-energy e + e − collider. The ratio R K is measured in five bins of dilepton invariant-mass-squared ( q 2 ): q 2 ∈ (0 . 1 , 4 . 0) , (4 . 00 , 8 . 12) , (1 . 0 , 6 . 0), (10 . 2 , 12 . 8) and ( > 14 . 18) GeV 2 /c 4 , along with the whole q 2 region. The R K value for q 2 ∈ (1 . 0 , 6 . 0) GeV 2 /c 4 is $$ {1.03}_{-0.24}^{+0.28} $$ 1.03 − 0.24 + 0.28 ± 0 . 01. The first and second uncertainties listed are statistical and systematic, respectively. All results for R K are consistent with Standard Model predictions. We also measure CP -averaged isospin asymmetries in the same q 2 bins. The results are consistent with a null asymmetry, with the largest difference of 2.6 standard deviations occurring for the q 2 ∈ (1 . 0 , 6 . 0) GeV 2 /c 4 bin in the mode with muon final states. The measured differential branching fractions, $$ d\mathrm{\mathcal{B}} $$ d ℬ /dq 2 , are consistent with theoretical predictions for charged B decays, while the corresponding values are below the expectations for neutral B decays. We have also searched for lepton-flavor-violating B → Kμ ± e ∓ decays and set 90% confidence-level upper limits on the branching fraction in the range of 10 − 8 for B + → K + μ ± e ∓ , and B 0 → K 0 μ ± e ∓ modes.