## 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