Dear all, I really want to know the answer of the following question. I would appreciate any help.

Assume H is a separable Hilbert space, is it possible to find two nests N1, N2 such that the intersection of Alg(N1) and Alg(N2) contains only scalars? If dimH is finite, then it is quiet easy to answer the similar question negatively, but I have no idea how to approach the question if dimH is infinite. I hope someone could provide some hint.

Thanks in advance!