Vol. 81, No. 2, 1979

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Extended weak-Dirichlet algebras

Takahiko Nakazi

Vol. 81 (1979), No. 2, 493–513

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 E to some sub σ-algebra 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.

Mathematical Subject Classification 2000
Primary: 46J15
Received: 17 August 1978
Published: 1 April 1979
Takahiko Nakazi
Hokusei Gakuen University
2-3-1, Ohyachi-Nishi
Atsubetsu-ku Sapporo 004-8631