Test of lepton flavor universality and search for lepton flavor violation in $B \to K \ell\ell$ decays.

We present measurements of the branching fractions for the decays $B\to K \mu^{+}\mu^{-}$ and $B\to K e^{+}e^{-}$, and their ratio ($R_{K}$), using a data sample of 711 $fb^{-1}$ that contains $772 \times 10^{6}$ $B\bar{B}$ events. The data were collected at the $\Upsilon(4S)$ 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 ($q^{2}$): $q^{2} \in (0.1, 4.0), (4.0, 8.12), (1.0, 6.0)$, $(10.2, 12.8)$ and ($>14.18) GeV^{2}/c^{4}$, along with the whole $q^2$ region. The $R_{K}$ value for $q^{2} \in (1.0, 6.0) GeV^{2}/c^{4}$ is $1.03^{+0.28}_{-0.24} \pm 0.01$. 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 $C\!P$-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 $q^{2}\in(1.0,6.0) GeV^{2}/c^{4}$ bin in the mode with muon final states. The measured differential branching fractions, ${d\cal B}/{dq^{2}}$, 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 $B \rightarrow K\mu^{\pm}e^{\mp}$ decays and set $90\%$ confidence-level upper limits on the branching fraction in the range of $10^{-8}$ for $B^{+} \rightarrow K^{+}\mu^{\pm}e^{\mp}$, and $B^{0} \rightarrow K^{0}\mu^{\pm}e^{\mp}$ modes.