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Kind of, but JPEG converts image data to its own internal 3 came channel colour space before applying DCT. It is not compressing the R, G and B channels of most images. So a multichannel compression is not just compressing each channel separately.
Yeah, jpeg converts to lab (or something similar, I think). But the dimensions are the same: one channel for lightness, and then a number of channels one less than the total number of sampled frequencies to capture the rest of the color space.