Complete algebraic positivity of essential Herz–Schur multipliers on the essential convolution algebra

Determine whether, for every étale groupoid G and every function ξ in the convolution *-algebra A_c(G) of compactly supported functions continuous on a bisection, the Herz–Schur multiplier m_ϕ^{E_ℒ}: A_c^{ess}(G) → A_c^{ess}(G) induced by ϕ = ξ^* * ξ is completely algebraically positive. Here A_c^{ess}(G) denotes the essential convolution algebra obtained by quotienting A_c(G) by the ideal of elements whose support is contained in the set of dangerous arrows, and E_ℒ denotes the essential conditional expectation used to define A_c^{ess}(G).

Background

In Section 4 the authors construct Herz–Schur multipliers m_ϕ associated to functions ϕ = ξ^* * ξ with ξ in A_c(G), and show these induce completely positive (c.c.p.) maps on the full, reduced, and maximal essential C*-algebras. To carry out Stinespring-type arguments in the essential setting, they verify a form of complete algebraic positivity for m_ϕ^{E_ℒ} when viewed as mapping into the Borel-ess quotient algebra B_c^{ess}(G) = B_c(G)/M_c(G).

They note that the image of m_ϕ^{E_ℒ} actually lies in the essential convolution algebra A_c^{ess}(G)⊆B_c^{ess}(G), but they do not know whether m_ϕ^{E_ℒ} is completely algebraically positive as a map A_c^{ess}(G)→A_c^{ess}(G). Establishing this would strengthen the technical framework used in the proofs relating essential amenability and nuclearity.