Catalyzes the formation of 5-oxoproline from gamma-glutamyl dipeptides and plays a significant role in glutathione (GSH) homeostasis. Converts GSH to 5-oxoproline and cysteine-glycine (Cys-Gly) dipeptide in vitro. Possesses low activity towards gamma-glutamyl-L-alanine. Has no activity towards gamma-glutamyl-L-cysteine. an alpha-(gamma-L-glutamyl)-L-amino acid = 5-oxo-L-proline + an L-alpha-amino acid Binds 2 Mn(2+) ions per subunit. Belongs to the gamma-glutamylcyclotransferase family.