Universal ${\mathrm{C}}^{\ast }$-algebras with the Local Lifting Property

The Local Lifting Property (LLP) is a localized version of projectivity for completely positive maps between ${\mathrm{C}}^{\ast }$-algebras. Outside of the nuclear case, very few ${\mathrm{C}}^{\ast }$-algebras are known to have the LLP. In this article, we show that the LLP holds for the algebraic contraction ${\mathrm{C}}^{\ast }$-algebras introduced by Hadwin and further studied by Loring and Shulman. We also show that the universal Pythagorean ${\mathrm{C}}^{\ast }$-algebras introduced by Brothier and Jones have the Lifting Property.