Ergodicity, memory loss, and directional stability on von Neumann algebras

Abstract

For a deterministic sequence of normal unital completely positive maps on a von Neumann algebra, convergence of normal states, loss of memory, and convergence of Cesàro averages are different properties. This paper separates these properties in the predual norm and identifies the precise role of composition order. Uniform loss of memory for right-appended products on the predual implies uniform convergence to a normal state without any compactness or trace assumption; the convergence rate is bounded by the image diameter. In contrast, on every infinite-dimensional $σ$-finite von Neumann algebra there are right-appended products with pointwise loss of memory but no convergent normal-state trajectory, even though all nontrivial state images equal a fixed compact line segment. Consequently, finite dimensionality is characterized by the universal implication from pointwise backward memory loss to mixing. An attained normal-functional minorization coefficient extends the common-lower-bound construction for matrix channels to arbitrary von Neumann algebras and yields quantitative contraction and unique normal entrance laws. Further results distinguish pointwise from uniform relaxation, Cesàro convergence from mixing, fixed-start from time-uniform contraction, and stationary from moving asymptotic states. As an application, inhomogeneous transfer states on spin chains, including matrix product states, are shown to have a unique thermodynamic limit, uniformly in the boundary condition, with an arbitrary von Neumann algebra as bond algebra; boundary dependence and correlations are controlled by the same product coefficients. All counterexamples are explicit and extend to algebras of types I, II, and III.