Let (X,𝒜,m) be a
probability measure space and A a subalgebra of L∞(m), containing the constant
functions. Srinivasan and Wang defined A to be a weak-*Dirichlet algebra if A +Ā
(the complex conjugate) is weak-*dense in L∞(m) and the integral is multiplicative
on A, ∫fg dm =∫f dm∫g dm for f,g ∈ A. In this paper the notion of extended
weak-*Dirichlet algebra is introduced; A is an extended weak-*Dirichlet algebra if
A +Ā is weak-*dense in L∞(m) and if the conditional expectation Eℬ to some sub
σ-algebra ℬ is multiplicative on A. Then most of important theorems proved for
weak-*Dirichlet algebras are generalized in the context of extended weak-*Dirichlet
algebras, for instance, Szegő’s theorem and Beuring’s theorem. Besides, our
approach will yield several theorems which were not known even for weak-*Dirichlet
algebras.
