2025/07/30 by Huang, Linzhe, Ma, Minghui
#46K50 #46L10 #47B47 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2507.22508
This paper investigates derivations of the free semigroupoid algebra \mathfrakLG of a countable or uncountable directed graph G and its norm-closed version, the tensor algebra AG. We first prove a weak Dixmier approximation theorem for \mathfrakLG when G is strongly connected. Using the theorem, we show that if every connected component of G is strongly connected, then every bounded derivation δ from AG into \mathfrakLG is of the form δ=δT for some T∈\mathfrakLG with ‖T‖\leqslant‖δ‖. For any finite directed graph G, we also show that the first cohomology group H1(AG,\mathfrakLG) vanishes if and only if every connected component of G is either strongly connected or a fruit tree. To handle infinite directed graphs, we introduce the alternating number and propose \Crefconj intro-in-tree. Suppose every connected component of G is not strongly connected. We show that if every bounded derivation from AG into \mathfrakLG is inner, then every connected component of G is a generalized fruit tree and the alternating number A(G) of G is finite. The converse is also true if the conjecture holds. Finally, we provide some examples of free semigroupoid algebras together with their nontrivial first cohomology groups.