Towards new background independent representations for loop quantum gravity

Recently, uniqueness theorems were constructed for the representation used in loop quantum gravity. We explore the existence of alternate representations by weakening the assumptions of the Lewandowski-Okolow-Sahlmann-Thiemann (LOST) uniqueness theorem. The weakened assumptions seem physically reasonable and retain the key requirement of explicit background independence. For simplicity, we restrict attention to the case of gauge group U(1)