# Torres Ayala, Francisco (2012-07). On Primitivity and the Unital Full Free Product of Finite Dimensional C*-algebras. Doctoral Dissertation. Thesis

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### abstract

• A C*-algebra is called primitive if it admits a *-representation that is both faithful and irreducible. Thus the simplest examples are matrix algebras. The main objective of this work is to classify unital full free products of finite dimensional C*-algebras that are primitive. We prove that given two nontrivial finite dimensional C*-algebras, A1 /= C, A2 /= C, the unital C*-algebra full free product A = A1 * A2 is primitive except when A1 = C^2 = A2. Roughly speaking, we first show that, except for trivial cases and the case A1 = C^2 = A2, there is an abundance of irreducible finite dimensional *-representations of A. The latter is accomplished by taking advantage of the structure of Lie group of the unitary operators in a finite dimensional Hilbert space. Later, by means of a sequence of approximations and Kaplansky?s density theorem we construct an irreducible and faithful {representation of A. We want to emphasize the fact that unital full free products of C*-algebras are highly abstract objects hence finding an irreducible *-{representation that is faithfully is an amazing fact. The dissertation is divided as follows. Chapter I gives an introduction, basic definitions and examples. Chapter II recalls some facts about *-automorphisms of finite dimensional C -algebras. Chapter III is fully devoted to prove Theorem III.6 which is about perturbing a pair of proper unital C*-subalgebras of a matrix algebra in such a way that they have trivial intersection. Theorem III.6 is the cornerstone for the rest of the results in this work. Lastly, Chapter IV contains the proof of the main theorem about primitivity and some consequences.

• July 2012