Date of Award
Spring 2023
Project Type
Dissertation
Program or Major
Mathematics
Degree Name
Doctor of Philosophy
First Advisor
Junhao Shen
Second Advisor
Rita Hibschweiler
Third Advisor
Donald Hadwin
Abstract
Wigner’s theorem, an important result in quantum mechanics, shows that a transitionprobability preserving bijection on the set of rank-one projections on a Hilbert space H extends to a Jordan ∗-isomorphism ρ on the algebra of all bounded linear operators on H. In this work we consider generalizations of Wigner’s theorem within the context of von Neumann algebras. Let M and N be two semi-finite von Neumann algebras with faithful normal semi-finite tracial weights τM and τN, respectively. Suppose that M has no direct summand of type I2 and is either diffuse or atomic such that every minimal projection in M has trace 1. Let c ∈ (0,τM(IM)/2), Pc(M, τM) = {P ∈ P(M) | τM(P) = c}, and 0 < p < ∞. In this work we prove generalizations of Wigner’s theorem along two main lines. First, we show that every ortho-isomorphism φ : Pc(M, τM) → Pc(M, τM) can be extended to a Jordan ∗-isomorphism ρ : M → M. Second, we show that if φ : Pc(M, τM) → Pc(N , τN) acts as an Lp-isometry on commuting pairs of projections, then φ extends to a trace-preserving Jordan ∗-homomorphism ρ : M → N when τM(IM) = τN(IN ) < ∞ or 0 < p ≤ 2.
Recommended Citation
Gent, Devin Daley, "Wigner's Theorem for Semi-finite von Neumann Algebras" (2023). Doctoral Dissertations. 2738.
https://scholars.unh.edu/dissertation/2738