Dalitz plot analysis of the decay<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:msup><mml:mi>K</mml:mi><mml:mo>±</mml:mo></mml:msup><mml:msup><mml:mi>K</mml:mi><mml:mo>∓</mml:mo></mml:msup></mml:math>

We analyze the three-body charmless decay ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{K}^{\ifmmode\pm\else\textpm\fi{}}{K}^{\ifmmode\pm\else\textpm\fi{}}{K}^{\ensuremath{\mp}}$ using a sample of $226.0\ifmmode\pm\else\textpm\fi{}2.5$ million $B\overline{B}$ pairs collected by the BABAR detector. We measure the total branching fraction and $CP$ asymmetry to be $\mathcal{B}=(35.2\ifmmode\pm\else\textpm\fi{}0.9\ifmmode\pm\else\textpm\fi{}1.6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and ${A}_{CP}=(\ensuremath{-}1.7\ifmmode\pm\else\textpm\fi{}2.6\ifmmode\pm\else\textpm\fi{}1.5)%$. We fit the Dalitz plot distribution using an isobar model and measure the magnitudes and phases of the decay coefficients. We find no evidence of $CP$ violation for the individual components of the isobar model. The decay dynamics is dominated by the ${K}^{+}{K}^{\ensuremath{-}}$ $S$-wave, for which we perform a partial-wave analysis in the region $m({K}^{+}{K}^{\ensuremath{-}})<2\text{ }\text{ }\mathrm{GeV}/{c}^{2}$. Significant production of the ${f}_{0}(980)$ resonance, and of a spin zero state near $1.55\text{ }\text{ }\mathrm{GeV}/{c}^{2}$ are required in the isobar model description of the data. The partial-wave analysis supports this observation.

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