# States, spectra, and numerical radii in operator algebras

** Speaker:** John D. Trout, Dartmouth College

** Date:** September 28, 2023

** Abstract:** We will discuss the relationship between spectra of elements and states in unital Banach and C*-algebras, and how these relate to the numerical radii of elements. For example, we will discuss elementary proofs that every member of the spectrum of any element in a unital Banach algebra is realized by some state (i.e., unital linear functional of norm one) on the algebra. Then, for various algebras, we will show how to explicitly construct the states that realize these spectral elements. We will then strengthen this result using the holomorphic functional calculus.