The e-Property for Quantum Markov Semigroups

Let $(\mathcal{T}_t)_{t\geqslant0}$ be a quantum Markov semigroup on a von Neumann algebra and let $\mathcal{A}$ be an invariant Feller $C^*$-algebra. We formulate the $e$-property on the full state space and show that its quantitative defect is equivalent, within an optimal factor two, to the Hausdorff measure of non-compactness of observable orbits. We develop Cesàro and Lip-norm criteria and a generalised Doeblin minorisation by a state-space contracting positive map, including finite-dimensional and measure-prepare factorisations. When every observable has the $e$-property, the Jacobs–de Leeuw–Glicksberg decomposition identifies the reversible Choi–Effros algebra and the norm-stable sector. Expected and decoherence-free subalgebras transfer the property under norm decay of the complementary dynamics, including infinite-dimensional ultracontractive and finite-index Pimsner–Popa regimes. Finally, we establish the full comparison hierarchy among ordinary, eventual, Cesàro and pointwise eventual equicontinuity notions, reinforced by matrix Jamison and homoclinic flow counterexamples. Projection-face criteria are treated in a companion paper.