Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph C⁎-algebras

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We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra LK(E) of an arbitrary graph E with coefficients in a field K is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every K-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a C⁎-algebra of a countable graph is isomorphic to the C⁎-algebra of an ample groupoid.