Pressure-induced variations of $^{27}$Al NMR spectra of CeAl$_3$ indicate significant changes in the ground-state characteristics of this prototypical heavy-electron compound. Previously reported magnetic and electronic inhomogeneities at ambient pressure and very low temperatures are removed with external pressures exceeding 1.2 kbar. The spectra and results of corresponding measurements of the NMR spin-lattice relaxation rates indicate a pressure-induced emergence of a simple paramagnetic state involving electrons with moderately enhanced masses and no magnetic order above 65 mK.
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