The appearance of color stimuli with luminance levels beyond the diffuse white is gaining importance due to the popularity of high dynamic range (HDR) displays. Past work on color appearance of stimuli, color appearance models, and uniform color spaces mainly focused on the stimuli with luminance levels below the diffuse white, which were produced using surface color samples or conventional standard dynamic range (SDR) displays. In this study, we focused on the perception of white appearance for stimuli with luminance beyond the diffuse white. Human observers adjusted the color appearance of a stimulus to the whitest under different adapting conditions, including a dark condition and 12 illuminated conditions. It was found that the chromaticities for producing the white appearance under the dark condition were generally similar to those under the 6500 K conditions, regardless of the adapting luminance levels. In comparison to a recent study focusing on the stimuli with luminance below the diffuse white, the perception of white under the conditions with the adapting CCT levels of 2700, 3500, and 5000 K was significantly affected by the lightness level of the stimulus, which cannot be accurately characterized by CAM02-UCS. The results can be used for reproducing white appearance for highlights in HDR scenes. Further investigations on uniform color spaces for characterizing stimuli with luminance beyond the diffuse white are urgently needed for processing and displaying HDR images.