MULTIPLICATIVITY FACTORS FOR FUNCTION SEMINORMS

Let (X, A, mu) be a measure space, let rho be a function seminorm on M = IU(X, A, mu) the algebra of measurable functions on X, and let L(rh o) be the space {f is an element of M : rho(f) < infinity}. We prove t hat if rho is sigma-subadditive, then L(rho) is an algebra if and only if L(rho) is contained in L(infinity) + Kernel of rho. Further, we ob tain the best (least) multiplicativity factor for rho. In the case tha t rho is a function norm we improve a result proved previously, which says that the best multiplicativity factor for rho is determined by su p{parallel to f parallel to(infinity) : f is an element of L(rho), rho (f) less than or equal to 1}, Also, we introduce another function semi norm associate to rho. (C) 1995 Academic Press, Inc.