| It is first proven that the adjoint of a minimal projection in ℓ13 (resp. ℓinfinity3 ) is minimal in ℓinfinity3 (resp. ℓ13 ). Furthermore, it is shown rank-two minimal projections in ℓ13 and ℓinfinity3 are unique.;Next a characterization of all Rank-two minimal projections in ℓinfinity4 whose adjoints are minimal in ℓ14 is given. It is then proven that the lebesgue function is constant on all proper rank-two minimal projections in ℓ1n . However, it turns out, P minimal in ℓinfinity4 with the lebesgue function constant on P* is not a sufficient condition to guarantee the minimality of P* in ℓ14 . It is next demonstrated that minimal projections onto a certain class of orthogonal subspaces must have a particularly simple form. Moreover, it is shown precisely when the adjoint of a minimal projection in ℓk4 or ℓk6 onto these subspaces is also minimal for k = 1 or infinity. After which we provide an example of a subspace, H, of ℓ1n for which the adjoint of every minimal projection onto H is not minimal in ℓinfinityn .;Continuing we define the notion of a global minimal projection. After which, it is proven that certain symmetry conditions on a subspace are sufficient to guarantee the existence of a global minimal projection onto it. Furthermore, we classify all rank-two global minimal projections in the spaces ℓqn for q ≥ 1 and n = 3, 4, and 6. As an application of global minimal projections, we provide a new proof that for p ≥ 1 and n ≥ 3, ℓqn is a Hilbert space iff p=2. Lastly, we create some computer programs using mathematica that help in the computation of minimal projections in the space ℓ1n for n=4 and 5. |
