Characterization of subdiagonal algebras

Abstract

Abstract. Let M be a finite von Neumann algebra with a faithful normal tracial state τ, and let A be a tracial subalgebra of M. We show that A has L p -factorization (1 ≤ p < ∞) if and only if A is a subdiagonal algebra. Also, we obtain some characterizations of subdiagonal algebras. IntroductionLet M be a finite von Neumann algebra with a faithful normal tracial state τ. In [1], Arveson introduced the notion of finite, maximal, subdiagonal algebras A of M, as noncommutative analogues of weak-* Dirichlet algebras. Subsequently several authors studied the (noncommutative) H p -spaces associated with such algebras ([9, 11, 12, 13, 14, 16, 17, 18] [3,4,5], among other things, Blecher and Labuschagne studied tracial subalgebras of M and gave several characterizations of subdiagonal algebras. They proved that if a tracial subalgebra A has L ∞ -factorization, then A is a subdiagonal algebra. We will consider the L p -factorization (0 < p < ∞) property of tracial subalgebras. This paper is organized as follows. Section 1 contains some preliminary definitions. In section 2, we prove that if a tracial subalgebra A has L p -factorization (1 ≤ p < ∞), then A is a subdiagonal algebra. In section 3, we consider tracial subalgebras, which satisfy L 2 -density. We show that if a tracial subalgebra A has L p -factorization (0 < p < 1) and satisfies L 2 -density, then A is a subdiagonal algebra. PreliminariesThroughout this paper, we denote by M a finite von Neumann algebra on a Hilbert space H with a faithful normal tracial state τ . For 0 [7,20]).

Factorization properties of subdiagonal algebras

“…In [4] and [5], among other things, Blecher and Labuschagne proved that a tracial subalgebra [4,Definition 5.5] A has an L ∞ -factorization if and only if A is a subdiagonal algebra. In [3] it was shown that if a tracial subalgebra has an L p -factorization (0 < p < ∞), then it is a subdiagonal algebra.…”