arXiv:1906.00841v2 [math.OA] 11 Jul 2021 Noname manuscript No. (will be inserted by the editor) A Beurling-Blecher-Labuschagne type theorem for Haagerup noncommutative L p spaces Turdebek N. Bekjan · Madi Raikhan Received: date / Accepted: date Abstract Let M be a σ-finite von Neumann algebra, equipped with a normal faithful state ϕ, and let A be maximal subdiagonal subalgebra of M and 1 ≤ p< ∞. We prove a Beurling-Blecher- Labuschagne type theorem for A-invariant subspaces of Haagerup noncommutative L p (A) and give a characterization of outer operators in Haagerup noncommutative H p -spaces associated with A. Keywords subdiagonal algebras, Beurling’s theorem, invariant subspace, outer operator, Haagerup noncommutative H p -space Mathematics Subject Classification (2010) 46L52 · 47L05 1 Introduction Arveson introduced his notion of subdiagonal subalgebras of von Neumann algebras (see [1]), in effect, subdiagonal algebras are the noncommutative analogue of weak* Dirichlet algebras (for the definition of weak* Dirichlet algebras see [23]). For the finite and semi-finite case, most results on the classical Hardy spaces on the torus have been established in this noncommutative setting. We refer to [1,2,3,4,7,8,6,9,20,21] (see also [9] for more historical references). It is natural to consider the case of σ-finite von Neumann algebras. But, the transition from finite or semifinite to σ-finite von Neumann algebras is not trival, need some new techniques and some changes. For some results for this case, see [5,12,13,14,17,26]. Let M be a finite von Neumann and A be its Arveson’s maximal subdiagonal subalgebras. In [6], Blecher and Labuschagne extended the classical Beurling’s theorem to describe closed A- invariant subspaces in noncommutative space L p (M) with 1 ≤ p ≤∞. Sager [21] extended the work of Blecher and Labuschagne from a finite von Neumann algebra to semifinite von Neumann algebras, proved a Beurling-Blecher-Labuschagne theorem for A-invariant spaces of L p (M) when 0 <p ≤∞. The Beurling theorem has been generalized to the setting of unitarily invariant norms on finite and semifinite von Neumann algebras (see [4], [10], [22]). When A is subdiagonal subalgebra of σ-finite von Neumann M, Labu- schagne [17] showed that a Beurling type theory of invariant subspaces of noncommutative H 2 - spaces holds true. A motivation for this paper is to extend the result in [17] to the setting of the Haagerup noncommutative L p -spaces for 1 ≤ p< ∞. T.N. Bekjan is partially supported by NSFC grant No.11771372, M. Raikhan is partially supported by project AP05131557 of the Science Committee of Ministry of Education and Science of the Republic of Kazakhstan. T. N. Bekjan College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China. E-mail: bekjant@yahoo.com M. Raikhan Astana IT University, Nur-Sultan 010000, Kazakhstan. E-mail: madi.raikhan@astanait.edu.kz