  We study several trace operators and spaces that are related to the
bi-Laplacian. They are motivated by the development of ultraweak formulations
for the bi-Laplace equation with homogeneous Dirichlet condition, but are also
relevant to describe conformity of mixed approximations.
  Our aim is to have well-posed (ultraweak) formulations that assume low
regularity, under the condition of an $L_2$ right-hand side function. We pursue
two ways of defining traces and corresponding integration-by-parts formulas. In
one case one obtains a non-closed space. This can be fixed by switching to the
Kirchhoff-Love traces from [F\"uhrer, Heuer, Niemi, An ultraweak formulation of
the Kirchhoff-Love plate bending model and DPG approximation, Math. Comp., 88
(2019)]. Using different combinations of trace operators we obtain two
well-posed formulations. For both of them we report on numerical experiments
with the DPG method and optimal test functions.
  In this paper we consider two and three space dimensions. However, with the
exception of a given counterexample in an appendix (related to the
non-closedness of a trace space), our analysis applies to any space dimension
larger than or equal to two.
