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Measurement of branching fractions and rate asymmetries in the rare decays<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>*</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math>
In a sample of $471\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ events collected with the BABAR detector at the PEP-II ${e}^{+}{e}^{\ensuremath{-}}$ collider we study the rare decays $B\ensuremath{\rightarrow}{K}^{(*)}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$, where ${\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ is either ${e}^{+}{e}^{\ensuremath{-}}$ or ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$. We report results on partial branching fractions and isospin asymmetries in seven bins of dilepton mass-squared. We further present $CP$ …