Derivations for a Class of Matrix Function Algebras

We study a class of matrix function algebras, here denoted T (Cn). We introduce a notion of point derivations, and classify the point derivations for certain finite dimensional representations of T (Cn). We use point derivations and information about n×n matrices to show that every T (Cn)-valued derivation on T (Cn) is inner. Certain matrix function algebras arise in some standard constructions in the theory of non-selfadjoint operator algebras. They have been studied as semicrossed product operator algebras, see in particular [1] and [6]. More recently this same class of algebras have been realized as directed graph operator algebras, see [12, Example 6.5]. Directed graph operator algebras have been studied in relation to the standard commutative example, A(D), the algebra of holomorphic functions in the unit disk with continuous extensions to T. For example the automorphism groups of certain graph operator algebras have connections to the automorphism group of A(D), see [1] when the directed graph is a cycle and [5] when the directed graph has a single vertex. Another example is the description of the ideal theory for directed graph operator algebras given in [10]. In this paper we extend descriptions of point derivations of the disk algebra, see [3, Section 1.6] or [2], to a notion of point derivations on the directed graph operator algebras coming from cycles. We exploit the structure of these directed graph operator algebras as matrix function algebras to describe when point derivations occur. In the first section we define our algebras and establish some notation. In the second section we define a notion of point derivation and prove some preliminary results. In particular, we introduce inner point derivations and look for ways to tell whether a point derivation is inner. In the third section we apply these results to describe the point derivations of the matrix function algebras. We are able to classify 2000 Mathematics Subject Classification. 47L75, 46H35.