What is a double star-product?

Abstract: Double Poisson brackets, introduced by M. Van den Bergh in 2004, are noncommutative analogs of the usual Poisson brackets in the sense of the Kontsevich-Rosenberg principle: they induce Poisson structures on the space of $N$-dimensional representations $\operatorname{Rep}_N(A)$ of an associative algebra $A$ for any $N$. The problem of deformation quantization of double Poisson brackets was raised by D. Calaque in 2010, and had remained open since then. In this paper, we address this problem by answering the question in the title. We present a structure on $A$ that induces a star-product under the representation functor and, therefore, according to the Kontsevich-Rosenberg principle, can be viewed as an analog of star-products in noncommutative geometry. We also provide an explicit example for $A=\Bbbk\langle x_1,\ldots,x_d\rangle$ and prove a double formality theorem in this case. Along the way, we invert the Kontsevich-Rosenberg principle by introducing a notion of double algebra over an arbitrary operad.