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The Lab color space not only exceeds monitor gamut (as do AdobeRGB and ProPhotoRGB in most cases), it far exceeds even the RGB and CMYK gamuts.
One of the most important attributes of the Lab model is device independence. This means that the colors are defined independent of their nature of creation or the device they are displayed on. The Lab color space is used when graphics for print have to be converted from RGB to CMYK, as the Lab gamut includes both the RGB and CMYK gamut.
The huge space within Lab offers a safe conversion or interchange waypoint for a lot of color management tasks.
There are some powerful tricks you can do within Lab that you can't do elsewhere, too - for example, sharpening or changing contrast in the lightness channel without affecting the colors. If you’re interested in exploring some interesting possibilities, see Photoshop LAB Color: The Canyon Conundrum and other adventures in the Most Powerful Colorspace, by Dan Margulis.
- I get this point ->
sharpening or changing contrast in the lightness channel without affecting the colors
- I am not still clear about this one ->
If Lab color space exceeds monitor gamut, the colors that actually exceeds cannot be produced in monitors right? I couldn't really understand the sentence "It exceeds monitor gamut".
This has nothing to do with what a monitor can reproduce. It's about safely encoding data in a device-independent reference color space. Lab numbers are absolute, unambiguous. They refer to specific colors (whether a monitor can reproduce them or not).
That's why Lab is used in color management operations as a so-called Profile Connection Space. It's the firm ground you stand on when converting between all the other color spaces - the reference.
It's also useful because the luminance and chroma components are separate. There are countless scenarios where this is important.