Residual finite-dimensionality of ultragraph algebras via branching systems

We study residual finite-dimensionality for ultragraph algebras, both in the algebraic and in the C-star-algebraic settings.

We introduce graph-theoretic RFD conditions for ultragraphs, extending the conditions that characterize RFD graph C-star-algebras.

Using the boundary ultrapath branching system, we construct finite-dimensional branching-system representations associated to terminal boundary sets and no-exit cycles.

These representations are used to prove that, whenever an ultragraph satisfies the graph-theoretic RFD conditions, its ultragraph Leavitt path algebra LK(G) is RFD, for every field K, and its ultragraph C-star-algebra RFD.

For ultragraphs satisfying Condition (RFUM2), we prove converses in both settings.

The analytic converse uses the groupoid model and the density of periodic points, while the algebraic converse is proved directly by finite-dimensional linear algebra.

Thus, for RFUM2 ultragraphs, RFD of LK(G), RFD of C(G), and the graph-theoretic RFD conditions are equivalent.

This gives, in particular, a common combinatorial description linking the algebraic and analytic theories, recovers the graph C-start-algebra characterization, and yields an algebraic characterization for Leavitt path algebras of graphs.

We also construct an RFD ultragraph algebra which is genuinely outside the graph-algebra class in both settings.