Contributes to degradation of proteins cross-linked by transglutaminases by degrading the cross-link between a lysine and a glutamic acid residue. Catalyzes the formation of 5-oxo-L-proline from L-gamma-glutamyl-L-epsilon-lysine. Inactive with L-gamma-glutamyl-alpha-amino acid substrates such as L-gamma-glutamyl-L-alpha-cysteine and L-gamma-glutamyl-L-alpha-alanine. epsilon-(gamma-L-glutamyl)-L-lysine = 5-oxo-L-proline + L-lysine Monomer. Belongs to the gamma-glutamylcyclotransferase family.