2021/09/19 by Deguang Han, Qianfeng Hu, Han, Deguang +5 · 1 citation
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2109.09042
openalex publication_date 2021/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper concerns the dilations of Banach space operator-valued quantum measures. While the recently developed general dilation theory can lead to a projection (idempotent) valued dilation for any quantum measure over the projection lattice for a von Neumann algebra that dose not contain type I2 direct summand, such a dilation does not necessarily guarantee the preservation of countable additivity of the quantum measure. So it remain an open question whether every countably additive B(X)-valued quantum measure can be dilated to a countably additive projection-valued measure.The main purpose of this paper is to prove that such a dilation can be constructed if one of the following two conditions is satisfied: (i) the underling Banach space X = ℓp (1≤ p < 2) or it has Schur property, (ii) the quantum measure has bounded p-variation for some 1≤ p < ∞ . All of these were achieved by establishing a non-commutative version of a minimal dilation theory on the so-called elementary dilation space equipping with an appropriate dilation norm. In particular, the newly introduced p-variation norm on the elementary dilation space allows us to prove that every operator-valued quantum measure with bounded p-variation has a projection-valued quantum measure dilation that preserves the boundedness of the p-variation.