On primitive $3$-generated axial algebras of Jordan type

Abstract: Axial algebras of Jordan type $\eta$ are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta\not\in{0,1}$ is fixed, with restrictive multiplication rules. These properties generalize the Pierce decompositions for idempotents in Jordan algebras, where $\frac{1}{2}$ is replaced with $\eta$. In particular, Jordan algebras generated by idempotents are axial algebras of Jordan type $\frac{1}{2}$. If $\eta\neq\frac{1}{2}$ then it is known that axial algebras of Jordan type $\eta$ are factors of the so-called Matsuo algebras corresponding to 3-transposition groups. We call the generating idempotents {\it axes} and say that an axis is {\it primitive} if its adjoint operator has 1-dimensional 1-eigenspace. It is known that a subalgebra generated by two primitive axes has dimension at most three. The 3-generated case has been opened so far. We prove that any axial algebra of Jordan type generated by three primitive axes has dimension at most nine. If the dimension is nine and $\eta=\frac{1}{2}$ then we either show how to find a proper ideal in this algebra or prove that the algebra is isomorphic to certain Jordan matrix algebras.