Yes. I own a Virgin Mobile S3. I am currently running CM11 and a custom kernel (DKP) built for the Sprint S3.
Virgin Mobile S3 model #: SPH-L710
Sprint S3 model #: SPH-L710
Virgin Mobile use Sprint towers for service. I have 4G where I am and it's fast. Coverage isn't the best in multiple areas I travel to, but it's Sprint.
They do use a "backdoor", so to speak, to get to the Sprint network, so you can flash a .zip in recovery that a dev on another forum made to alter some APN and build.prop settings to allow you to have full access to the Virgin Mobile network when on a custom ROM.
Long story short, when you flash a d2lte ROM on your VM S3, it perceives itself to be a Sprint S3. If you need the zip to flash, I can direct you there.
Your "THANKS" are not necessary, but I would appreciate them when due.