The electronics related to the volume control is toast.
That player has the volume rocker on its right side towards the top of the unit; perhaps it became contaminated by something (moisture?), got subjected to an electrical surge or just became worn out from use. Without opening your unit up and inspecting/testing it, there's really no way to know for sure what the problem is.
Sony still services the A726, but given its age and that it's out of warranty it's probably not going to be cheap. Plus you'll have to pay to send it in to them, and even then there's no guarantee that it can be fixed.
I've had this mp3 player for about two years. About six months ago the sound stopped turning up, and will also turn down by itself. I've tried the reset button, which will set the sound back somewhere around 9. I've also tried the "restore factory settings" option, as well as reformatting it, and nothing has worked. I can't seem to find an answer to this question anywhere.