A note on positive bilinear maps - Davey, Aaron C. H.; Ivanescu, Cristian; Tcaciuc, Adi (My University, 2023)
This paper concerns positive maps between C∗-algebras, particularly when those positive maps are multilinear. We construct examples of positive bilinear maps that are not 2-positive, and therefore are not completely positive bilinear maps. Paulsen and Smith showed that completely bounded bilinear maps are in one-to-one correspondence with completely bounded linear maps. We show that a similar correspondence does not hold for positive bilinear and linear maps. In particular, we observe that a similar correspondence does not hold if we replace completely bounded with completely positive.
Keywords: C∗-algebras; multilinear; 2-positive; bounded bilinear maps; bounded linear maps; correspondence