Date of Award

Spring 2023

Project Type


Program or Major


Degree Name

Doctor of Philosophy

First Advisor

Junhao Shen

Second Advisor

Rita Hibschweiler

Third Advisor

Donald Hadwin


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.