Multiplicativity Factors for Function Seminorms

Let (X, A, μ) be a measure space, let ρ be a function seminorm on M = M(X, A, μ) the algebra of measurable functions on X, and let Lρ be the space {ƒ ∈ M : ρ(ƒ) < ∞}. We prove that if ρ is σ-subadditive, then Lρ is an algebra if and only if Lρ is contained in L∞ + Kernel of ρ. Further, we obtain the best (least) multiplicativity factor for ρ. In the case that ρ is a function norm we improve a result proved previously, which says that the best multiplicativity factor for ρ is determined by sup{||ƒ||∞ : ƒ ∈ Lρ, ρ(ƒ) ≤ 1}, Also, we introduce another function seminorm associate to ρ.