[section] [section] NONCOMMUTATIVE POINT DERIVATIONS FOR MATRIX FUNCTION ALGEBRAS

We discuss the notion of noncommutative point derivations for operator algebras. We look at the noncommutative point derivations for the quiver algebras, A(Cn) where Cn is the directed graph with n-vertices and n-edges forming a cycle. For the algebras Mn ⊗ A(D) and A(Cn) we classify when non-trivial point derivations can occur. We use the point derivations to show that every derivation on A(Cn) is inner. Non-selfadjoint operator algebras associated to directed graphs have recently undergone intensive study. They are an interesting class of operator algebras for which their underlying structure can be well understood. In this paper we continue this investigation for the class of graph operator algebras coming from n-cycles. We proceed in analogy to results concerning the disk algebra, the fundamental commutative graph operator algebra. For commutative function algebras, the notion of a point derivation can be connected intimately to analytic structure. There are examples where this connection breaks down, but for the disk algebra it is well known, see [2, Theorem 1] and [3, Section 1.6], that point derivations occur only at interior points of the maximal ideal space. In fact, a point of the maximal ideal space of the disk algebra is a peak point if and only if there are no nonzero point derivations at the point, see [3, Corollary 1.6.7]. In this paper we will extend these results to the graph algebras coming from n-cycles. Besides the interest in peak points, the analysis below is also useful in studying the homology groups associated to certain graph operator algebras. We will see that the first homology group of the graph operator algebras we discuss is trivial, in analogy with the same result for the disk algebra. 2000 Mathematics Subject Classification. 47L75, 46H35.