Extended weak-^∗Dirichlet algebras

Abstract

Let (X 9 J^9m) 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 + A (the complex conjugate) is weak-*dense in L°°(m) and the integral is multiplicative on A, \fgdm = \fdm \gβm for /, ge A. In this paper the notion of extended weak-*Dirichlet algebra is introduced; A is an extended weak-*Dirichlet algebra if A + A is weak-*dense in L'im) 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, Szegδ's theorem and Beuring's theorem. Besides, our approach will yield several theorems which were not known even for weak-*Dirichlet algebras.

Applications of the Fuglede-Kadison determinant: Szegö’s theorem and outers for noncommutative $H^p$

“…In 2004, the second author proved in [20] [21]). Another important historical remark is that the commutative case of most of the topics in our paper was settled in [23]. While this paper certainly gave us motivation to persevere in our endeavor, we follow completely different lines, and indeed the results work out rather differently too.…”

“…While this paper certainly gave us motivation to persevere in our endeavor, we follow completely different lines, and indeed the results work out rather differently too. In particular, the quantity τ (exp(Φ(log |f |))), which plays a central role in most of the results in [23], seems to us to be unrelated to outers or factorization in the noncommutative setting. In passing, we remark that numerical experiments do seem to confirm the existence of a Jensen inequality τ (exp(Φ(log |a|))) ≥ τ (|Φ(a)|) for subdiagonal algebras.…”

Szegő’s theorem on a bidisc

“…That is, Hr is an extended weak-* Dirichlet algebra with respect to ê?r [2]. If r is irrational then Hr is a weak-* Dirichlet algebra [5].…”

“…The following lemma and proposition are essentially known [2]. In this section we will improve the following known inequality:…”

“…In §3 we give expressions in terms of w of the L (w ff'w)-distances between 1 and subalgebras of L°° which contain &>Q properly. This follows from the theory of an abstract Hardy space [2]. In §4 we estimate S(w) from the above and the below by means which are defined using conditional expectations.…”