We introduce a new family of non-negative real-valued functions on a unital -algebra, i.e., for (Formula presented.) where and is an interpolation path of the symmetric mean. These functions are semi-norms as they satisfy the norm axioms, except for the triangle inequality. Special cases satisfying triangle inequality, and a complete equality characterization is also discussed. Various bounds and relationships will be established for this new family, with a connection to the existing literature in the algebra of all bounded linear operators on a Hilbert space.