We study Birkhoff-James orthogonality and its local symmetry in some sequence spaces namely ℓp, for 1 ≤ p ≤ ∞, p = 2, c, c0 and c00. Using the characterization of the local symmetry of Birkhoff-James orthogonality, we characterize isometries of each of these spaces onto itself and obtain the Banach-Lamperti theorem for onto operators on the sequence spaces.

We study Birkhoff-James orthogonality and its pointwise symmetry in commutative C * algebras, i.e., the space of all continuous functions defined on a locally compact Hausdorff space which vanish at infinity. We use this characterization to obtain the characterization of Birkhoff-James orthogonality on L ∞ space defined on any arbitrary measure space. We also do the same for the L p spaces for 1 ≤ p < ∞.

No abstract

We study the relationship between the point-wise symmetry of Birkhoff-James orthogonality and the geometry of the space of operators B(ℓ n ∞ , ℓ m 1 ). We show that any non-zero left-symmetric point in this space is a smooth point. We also show that for n ≥ 4, any unit norm right-symmetric point of this space is an extreme point of the closed unit ball. This marks the first step towards characterizing the extreme points of these unit balls and finding the Grothendieck constants G(m, n) using Birkhoff-James orthogonality techniques.

No abstract