Is there a box that can downconvert 1080p to 1080i

I still have a CRT RPTV in my living room and am looking into adding a 1080p projector in the near future. In other rooms I have 1080p plasmas and LCDs. I just ordered an HDMI matrix splitter to feed all of my rooms from the devices on my A/V rack. My CRT RPTV is my only device not capable of accepting a 1080p signal. Rather than configure my devices to display to the lowest common denominator (and I'm not sure my lcds [computer monitors] will accept 1080i), I would prefer to just set my devices to output 1080p and stick a box in between my CRT and the splitter to downconvert the 1080p signal to 1080i so it will accept it. Is there anything out there that will do this?

It's a 2003 or 2004 55" Mitsubishi Gold something or other. I am nearly certain that it won't accept 1080p, considering it won't even accept 720p. What I'm looking for is a converter box that will take a 1080p input signal and output 1080i.