Conditions for Primitivity of unital amalgamated full free products of finite dimensional C*-algebras

Abstract

We consider amalgamated unital full free products of the form $A_1{\ast }_D{A}_2$, where $A_1,A_2$ and $D$ are finite dimensional C*-algebras and there are faithful traces on $A_1$ and $A_2$ whose restrictions to $D$ agree. We provide several conditions on the matrices of partial multiplicities of the inclusions $D\hookrightarrow A_1$ and $D\hookrightarrow A_2$ that guarantee that the C*-algebra $A_1{\ast }_D{A}_2$ is primitive. If the ranks of the matrices of partial multiplicities are one, we prove that the algebra $A_1{\ast }_D{A}_2$ is primitive if and only if it has a trivial center.

arXiv comment40 pages. No essential changes in second submission. Some misspelled words were corrected and format was changed