Matrices Hermitian for an Absolute Norm

Let v be a (standardized) absolute norm on en. A matrix H in enn is called normHermitian jf the numerical range V(H) determined by v is real. Let :re be the set of all norm-Hermitians in en"' We determine an equivalence relation'" on {t, .•. , n} with the following property: Let HE en"' Then HE :re if and only if H is Hermitian and h,) = 0 if i + j. Let,l =.# + i:lC. Then.l is a subalgebra of en" and, for A e.1, Jl(A) equals the Euclidean numerical range and hence is convex. Let "f/" be the group of isometries for v, and let tpj = {exp(iH): H e2}. Then d/J is a nonnal subgroup of't" and 't" = dII~, where 9' is a group of permutation matrices.