Discontinuous subnorms

Let S be a subset of a finite-dimensional algebra over a field F, either R or C, so that S is closed under scalar multiplication. A real-valued functi on f defined on S, shall be called a subnorm if f(a) > 0 for all 0 not equa l a is an element of 6, and f(alphaa) = \ alpha \f(a) for all a is an eleme nt of S and alpha is an element of F. If in addition, S is closed under rai sing to powers, and f(a(m)) = f(a)(m) for all a is an element of S and m = 1, 2,3,..., then f shall be called a submodulus. Further, if S is closed un der multiplication, then a submodulus f shall be called a modulus if f(ab) = f(a)f(b) for all a, b is an element of S. Our main purpose in this paper is to construct discontinuous subnorms, submoduli and moduli, on the comple x numbers, the quaternions, and on suitable sets of matrices. In each of th ese cases we discuss the asymptotic behavior and stability properties of th e obtained objects.