We describe two kinds of regular invariant measures on the boundary path space ∂ E of a second countable topological graph E , which allows us to describe all extremal tracial weights on C*(E) which are not gauge-invariant. Using this description, we prove that all tracial weights on the C * -algebra C*(E) of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on C*(E) are gauge-invariant when C*(E) is simple and separable.
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Johannes Christensen (2024) studied this question.
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