Hard-material, flat-surfaced workpieces such as semiconductor wafers or sapphire disks are quickly attached with vacuum to a rotatable floating workpiece carrier. Fluid pressure in a sealed chamber applies uniform abrading pressure over the full abraded surface of the workpieces. A flexible diaphragm is used to form the sealed chamber and the carrier is rotationally driven by a lug-pin device. The floating carrier is horizontally restrained by a center-post device that provides rigid lateral support against abrading forces. Tilting of the floating carrier is provided by a spherical bearing. The abrading system can be operated at the very high abrading speeds used in high speed flat lapping with raised-island abrasive disks. Vacuum can also be applied to the sealed chamber to quickly move the workpiece away from the abrading surface. A mode of providing rigid parallel-surface abrading of a workpiece can be activated by simply applying vacuum to the sealed chamber.