Rennie will lead the joint venture with Bauer Media, which operates TV channels including 4Music, The Box and Smash Hits. He will lead the company's commercial strategy, working with advertisers and record labels.
At Global Radio since 2011, Rennie was in charge of all "non-core" revenue including the broadcaster's TV channels, bingo products, music and book publishing. He has previously worked at SeeSaw, Virgin Media, Warner Media and Channel 4.
Rose said: "Matt is a talented and highly respected music executive with a strong background in digital, broadcasting and music.
"We look forward to having him on board to take the Box TV business forward in this rapidly changing media and music landscape."
Katz left Box Television to become the director of Now TV at BSkyB.