We establish the Cauchy-Schwarz and Golden-Thompson inequalities for module operators, a generalization of a (noncommutative) conditional expectation, on a von Neumann algebra. We apply these inequalities to the Bennett inequality and an uncertainty relation, a generalization of the Schrödinger uncertainty relation, for conditional expectations.

Journal of Mathematical PhysicsJun 1, 20185 citations

Conditional Expectations on Jordan Algebras

C. Martin Edwards

NATO ASI series. Series B : PhysicsJan 1, 19863 citations

Theoretical and Mathematical Physics

Conditional expectations on von Neumann algebras and the causal independence of quantized fields

K. Yu. Dadashyan, Sergey S. Horujy

Published Feb 1, 1981• 0 citations

Publisher

DOI: 10.1007/bf01030843

Related papers

Conditional expectations on von Neumann algebras: A new approach

Stanley Gudder, Jean-Paul Marchand

Reports on Mathematical PhysicsDec 1, 197718 citations

Conditional expectation and commutativity in von Neumann algebras

Maciej Mączyński

International Journal of Theoretical PhysicsJun 1, 19773 citations

Sufficient positive maps between von Neumann algebras: Rényi divergences

Anna Jenčová, Lauritz van Luijk

We prove recovery theorems for $α$ - $z$ Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and $L^p$ -spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.We prove recovery theorems for $α$-$z$ Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and $L^p$-spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.

quant-pharXiv:2608.28510Aug 31, 2026

Different Types of Conditional Expectation and the Lüders—von Neumann Quantum Measurement

Gerd Niestegge

arXiv:1001.3842International Journal of Theoretical PhysicsJan 18, 20072 citations

Inequalities for positive module operators on von Neumann algebras

Byoung Jin Choi, Un Cig Ji, Yongdo Lim

Journal of Mathematical PhysicsJun 1, 20185 citations

Conditional Expectations on Jordan Algebras

C. Martin Edwards

NATO ASI series. Series B : PhysicsJan 1, 19863 citations

We prove recovery theorems for

α

z

Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and

L^p

-spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.