Quantum integrals (With an Appendix on the solution to Haagerup's weight problem)
Abstract
We solve two long outstanding problems here. We first solve in ZFC a problem that the first author and Weaver posed ten years ago: a state on a von Neumann algebra is countably additive on orthogonal projections if and only if it is sequentially weak* continuous, equivalently, sequentially normal. Such states may be considered the quantum (countably additive) probability measures, or rather their extension to a noncommutative integral. The principal tools are known ideas from direct integral theory from the 1970s. Indeed we prove a disintegration theorem for states on products of sigma-finite von Neumann algebras. We give several applications. For example we give a new variant of Gleason theorem, extending countably additive probability measures on projections to sequentially weak* continuous states when the algebra has no type I2 direct summand. We also establish normality and sequential normality criteria for weights, and give several other applications of our main result to weights. In forthcoming work we consider many applications to `quantum measure and integration theory', such as variants of Lebesgue's dominated convergence theorem for von Neumann algebras. The analytic tools and product-quotient construction developed here are used to prove in an Appendix by the first author that normality and sequential normality of arbitrary weights are determined by their restrictions to abelian von Neumann subalgebras, resolving Haagerup's Problem 1.11. (This may be moved to a companion paper.) That is, normality of weights is commutatively determined.
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BibTeX
@misc{blecher2026quantum,
title = {{Quantum integrals (With an Appendix on the solution to Haagerup's weight problem)}},
author = {David P. Blecher and Louis E. Labuschagne},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06771},
eprint = {2610.06771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06771},
}
RIS
TY - UNPB
TI - Quantum integrals (With an Appendix on the solution to Haagerup's weight problem)
AU - David P. Blecher
AU - Louis E. Labuschagne
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.06771
UR - https://arxiv.org/abs/2610.06771
AB - We solve two long outstanding problems here. We first solve in ZFC a problem that the first author and Weaver posed ten years ago: a state on a von Neumann algebra is countably additive on orthogonal projections if and only if it is sequentially weak* continuous, equivalently, sequentially normal. Such states may be considered the quantum (countably additive) probability measures, or rather their extension to a noncommutative integral. The principal tools are known ideas from direct integral theory from the 1970s. Indeed we prove a disintegration theorem for states on products of sigma-finite von Neumann algebras. We give several applications. For example we give a new variant of Gleason theorem, extending countably additive probability measures on projections to sequentially weak* continuous states when the algebra has no type I_2 direct summand. We also establish normality and sequential normality criteria for weights, and give several other applications of our main result to weights. In forthcoming work we consider many applications to `quantum measure and integration theory', such as variants of Lebesgue's dominated convergence theorem for von Neumann algebras. The analytic tools and product-quotient construction developed here are used to prove in an Appendix by the first author that normality and sequential normality of arbitrary weights are determined by their restrictions to abelian von Neumann subalgebras, resolving Haagerup's Problem 1.11. (This may be moved to a companion paper.) That is, normality of weights is commutatively determined.
ER -