Precise reference comparing operational and topological equivariant K-theory for toric varieties

Identify a precise published reference that establishes a formal comparison between the algebraic operational T-equivariant K-ring op K_T^0(X(Δ)) of a toric variety X(Δ) and the topological T_comp-equivariant K-ring K_{T_comp}^0(X(Δ)), such as an isomorphism or a canonical comparison map, thereby substantiating the assertion that the operational K-ring behaves well in relation to topological equivariant K-theory for toric varieties.

Background

The paper reviews algebraic and topological equivariant K-theory for varieties with group actions, emphasizing differences for singular varieties where natural isomorphisms between algebraic vector bundles and coherent sheaves, and between algebraic and topological K-theory, generally fail. To address this, the authors highlight the algebraic operational equivariant K-theory op K_T^0(X), which Anderson and Payne identified with piecewise Laurent polynomial functions for toric varieties.

In this discussion, the authors note that operational K-theory appears to be better behaved relative to the Grothendieck group of equivariant coherent sheaves and also relative to the topological equivariant K-ring. While they cite [4, Theorem 1.3] for the former relationship, they explicitly state they were unable to find a precise reference for the latter claim, indicating a concrete bibliographic gap in the literature for the comparison with topological equivariant K-theory.