All finite transitive graphs admit a self-adjoint free semigroupoid algebra

Abstract

In this paper we show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is B(H). This is accomplished through a new construction that reduces this problem to in-degree 2-regular graphs, which is then treated by applying the periodic Road Coloring Theorem of Béal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.2010 Mathematics Subject Classification. Primary: 47L55, 47L15, 05C20.

Structure of free semigroupoid algebras

“…This question had recently been resolved positively. In fact, in[20] it is shown that every non-cycle finite transitive graph has B(H) as a free semigroupoid algebra.…”