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High pressure effects on the superconducting transitions of beta-pyrochlore superconductors AOs2O6 (A = Cs, Rb, K) are studied by measuring resistivity under high pressures up to 10 GPa. The superconducting transition temperature Tc first increases with increasing pressure in all the compounds and then exhibits a broad maximum at 7.6 K (6 GPa), 8.2 K (2 GPa) and 10 K (0.6 GPa) for A = Cs, Rb and K, respectviely. Finally, the superconductivity is suppressed completely at a critical pressure near 7 GPa and 6 GPa for A = Rb and K and probably above 10 GPa for A = Cs. Characteristic changes in the coefficinet A of the T2 term in resistivity and residual resistivity are observed, both of which are synchronized with the corresponding change in Tc. It is suggested that electron correlations and certain quantum fluctuations play important roles in the occurrence or suppression of superconductivity in the beta-pyrochlore oxides.