This is the $24 Thanko Under the Armpit Cooling Device available from the Japan Trend Shop (so it'll cost $48 for two unless you only have a single problem-pit). It's a little three AAA battery powered fan than clips to short sleeves and blows air under your pits to keep them cool and dry. Or, if you're me, to spread the smell of a chili-cheese dog with onions. My pheromone game is on point.
Thanks to Mark, who bought two and informed me they also work great on balls.