What does path-connected property correspond to in Gelfand duality?

Question

A commutative C* algebra $A$ correspond to algebra of continuous function from locally compact Hausdorff space, where that space is the spectrum of $A$. Many properties of the space can be obtained from properties of $A$. I am wondering if spectrum of $A$ is path connected, what property of $A$ should have?

Answer 1

This is not an answer, but the following paper, Section 5, deals with a slightly weaker property, approximate path connectedness:

Hadwin, Don; Shulman, Tatiana, Tracial stability for $C^*$-algebras, Integral Equations Oper. Theory 90, No. 1, Paper No. 1, 35 p. (2018). ZBL1396.46045.