#### Date of Award

2018

#### Document Type

Open Access Dissertation

#### Department

Mathematics

#### First Advisor

Anton Schep

#### Abstract

In this dissertation we investigate the algebra numerical range defined by the Banach algebra of regular operators on a Dedekind complete complex Banach lattice, i.e., V (Lr(E), T) = {Φ(T) : Φ ∈ Lr(E)∗, ||Φ|| = 1 = Φ(I)}. For T in the center Z(E) of E we prove that V (Lr(E), T) = co(σ(T)). For T ⊥ I we prove that V (Lr(E), T) is a disk centered at the origin. We then consider the part of V (Lr(E), T) obtained by restricting ourselves to positive states Φ ∈ Lr(E)∗. In this case we show that we get a closed interval on the real line. Next we consider the problem of characterizing the linear maps on Lr(E) which preserve V (Lr(E), T). For this we first describe the regular states on Lr(E), in particular for the case E = `p(n) for 1 ≤ p ≤ ∞. This description allows us to show that any map Ψ on Lr(`p(n)) preserving V (Lr(`p(n)), T) for all T ∈ Lr(`p(n)) is of the form Ψ(T) = U ∗ (PtQTP) where U consists of elements of modulus 1, (∗) represents Hadamard multiplication, P is a permutation, and Q is a map that permutes off-diagonal entries of T. Furthermore, special conditions are given for Q for the cases p = 1, p = ∞ and p = 2. Finally, some extensions of these results to more general finite dimensional Banach lattices and infinite dimensional ` p’s are considered.

#### Recommended Citation

Sweeney, J. P.(2018). *States and the Numerical Range in the Regular Algebra.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/4954