Thomsen-Li's Theorem Revisited

TL;DR

This paper extends Thomsen-Li's Theorem to a broader class of Markov operators on C[0,1], showing they can be approximated by averages of homomorphisms that preserve specific subspaces, aiding in C*-algebra analysis.

Contribution

It provides a generalized version of Thomsen-Li's Theorem for subspace-preserving Markov operators on C[0,1], with implications for subhomogeneous C*-algebra constructions.

Findings

Approximation of Markov operators by homomorphisms in strong operator topology

Preservation of subspaces by averaged homomorphisms

Potential applications in constructing homomorphisms for subhomogeneous C*-algebras

Abstract

In this paper, we prove a general version of Thomsen-Li's Theorem--a Krein-Milman type theorem for C*-algebras. To be precise, for a Markov operator on $C[0,1]$ which preserves certain subspace of $C[0,1]$, we approximate it by an average of homomorphisms on $C[0,1]$ in the strong operator topology, additionally we require that this average also preserves the same subspace. Such results could be useful for constructing homomorphisms for subhomogeneous C*-algebras.