On order isomorphisms of locally parallel sets between C⁎-algebras: The case of matrix algebras

We introduce the notion of locally parallel sets of elements of $C^{⁎}$-algebras which generalizes that of norm attainment sets of continuous functions, and consider a preserver problem on order isomorphisms of locally parallel sets. It is shown that unitary multiplications of Jordan ⁎-isomorphisms give examples of order isomorphisms of locally parallel sets. This is achieved by analyzing the quasi-strong Birkhoff-James orthogonality, a new orthogonality relation between the original Birkhoff-James orthogonality and the strong Birkhoff-James orthogonality. Moreover, it is shown that every order isomorphism of locally parallel sets on the $C^{⁎}$-algebra of all $n\times n$ complex matrices is essentially implemented by two unitary matrices provided that $n\ge 3$.