Balls American women managed (7)
I believe the answer is:
oversaw
'managed' is the definition.
('oversee' can be a synonym of 'manage')
'balls american women' is the wordplay.
'balls' becomes 'overs' (an over is a sequence of balls bowled in cricket).
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'women' becomes 'w' (standard abbreviation).
'overs'+'a'+'w'='OVERSAW'
(Other definitions for oversaw that I've seen before include "watcher" , "Managed" , "What supervisor did" , "Supervised" .)