How do I make this picture white on transparent background and keep it's semitransparent parts -
(Background is already transparent).
Simply add a Color Overlay Layer Style and set that to white.
Layer > Layer Style > Color Overlay
choose the appropriate color you want to transform to alpha
Blue will make the background fully transparent, light-blue areas around the white element will be semi-transparent.
Dark red will make a semi-transparent blue background. But red elements in a picture will be made transparent too.
The effect you want to achieve is a bit unclear, as the image you posted did not have any transparency.