Coefficientwise formal associative deformations

Let $A$ be a commutative unital algebra over $\mathbb{R}$, with multiplication $m(f,g)=fg$. Write $A\mathrm{⟦}\mathrm{\hslash }\mathrm{⟧}$ for the real formal power-series module with coefficients in $A$; its underlying coefficient and Cauchy-product conventions are those of Formal power series over a commutative ring and the coefficient-extraction functional $[x^n]$. The $\mathrm{\hslash }$-adic topology is the coefficient-prefix topology of Order of a formal series, congruence modulo $x^N$, and the $x$-adic notions of convergence and Cauchy sequence: two series are close modulo ${\mathrm{\hslash }}^N$ when their coefficients agree in degrees below $N$.

A formal associative deformation of $A$ is specified by real-bilinear maps $B_r:A\times A\to A$ for $r\ge 1$, with $B_0:=m$, and the product $\left({\sum }_{i\ge 0}{\mathrm{\hslash }}^i{f}_i\right)\star \left({\sum }_{j\ge 0}{\mathrm{\hslash }}^j{g}_j\right)={\sum }_{N\ge 0}{\mathrm{\hslash }}^N{\sum }_{\begin{array}{c}i,j,r\ge 0\\ i+j+r=N\end{array}}{B}_r(f_i,g_j)\mathrm{.}$ For each fixed $N$ the inner sum is finite, so this formula defines a coefficientwise-continuous, $\mathbb{R}\mathrm{⟦}\mathrm{\hslash }\mathrm{⟧}$-bilinear product. It is required to be associative and unital with unit $1$. The unit requirement is equivalent to $B_r(1,f)=B_r(f,1)=0$ for $r\ge 1$. Reduction modulo $\mathrm{\hslash }$ is the original product $m$.

This definition alone asserts no differential-order condition, no specified first-order bracket, no convergence at a numerical value of $\mathrm{\hslash }$, and no existence or classification theorem.