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Reflexive homology and involutive Hochschild homology as equivariant Loday constructions

For associative rings with anti-involution several homology theories exists, for instance reflexive homology as studied by Graves and involutive Hochschild homology defined by Fernàndez-València and Giansiracusa. We prove that the corresponding homology groups can be identified with the homotopy groups of an equivariant Loday construction of the one-point compactification of the sign-representation evaluated at the trivial orbit, if we assume that $2$ is invertible and if the underlying abelian group of the ring is flat. We also show a relative version where we consider an associative $k$-algebra with an anti-involution where $k$ is an arbitrary ground ring.