arXiv subjects
Publications and source records attributed to Benton L. Duncan.
We use results on inclusions of free products and extensions of completely positive maps to determine the maximal $C^*$-envelope for upper triangular $3 \times 3$ matrices. We consider these same results in the context of larger upper triangular matrices and graph algebras associated to cycle graphs.
We use the Fock semicrossed product to define an operator algebra associated to a module over an integral domain. We consider the $C^*$-envelope of the semicrossed product, and then consider properties of these algebras as models for studying general semicrossed products.
Let $\sS$ be a countable, abelian semigroup of continuous surjections on a compact metric space $X$. Corresponding to this dynamical system we associate two operator algebras, the tensor algebra, and the semicrossed product. There is a unique smallest C$^*$-algebra into which an operator algebra is completely isometrically embedded, which is the C$^*$-envelope. We provide two distinct characterizations of the C$^*$-envelope of the tensor algebra; one developed in a general setting by Katsura, and the other using tools of projective and inductive limits, which gives the C$^*$-envelope as a crossed product C$^*$-algebra. We also study two natural classes of representations, the left regular representations and the orbit representations. The first is Shilov, and the second has a Shilov resolution.
We study operator algebras associated to integral domains. In particular, with respect to a set of natural identities we look at the possible nonselfadjoint operator algebras which encode the ring structure of an integral domain. We show that these algebras give a new class of examples of semicrossed products by discrete semigroups. We investigate the structure of these algebras together with a particular class of representations.
We investigate free products of finite dimensional $C^*$-algebras with amalgamation over diagonal subalgebras. We look to determine under what circumstances a given free product is exact and/or nuclear. In some cases we find a description of the algebra in terms of a more readily understood algebra.
We develop a notion of a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection. We associate to these generalized Cuntz-Krieger families a directed graph, with a coloring function on the edge set. We call such a directed graph an edge-colored directed graph. We then study the $C^*$-algebras and the non-selfadjoint operator algebras associated to edge-colored directed graphs. These algebras arise as free products of directed graph algebras with amalgamation. We then determine the $C^*$-envelopes for a large class of the non-selfadjoint algebras. Finally, we relate properties of the edge-colored directed graphs to properties of the associated $C^*$-algebra, including simplicity and nuclearity. Using the free product description of these algebras we investigate the $K$-theory of these algebras.
In analogy with the C*-algebra theory, we study variants appropriate to nonselfadjoint algebras of nuclearity, the local lifting property, exactness, and the weak expectation property. In addition, we study the relationships between these notions, and how they are connected with the classical C*-algebra theory through the use of C*-algebras generated by the algebra.
We show that for a class of operator algebras satisfying a natural condition the $C^*$-envelope of the universal free product of operator algebras $A_i$ is given by the free product of the $C^*$-envelopes of the $A_i$. We apply this theorem to, in special cases, the $C^*$-envelope of the semicrossed products for multivariable dynamics in terms of the single variable semicrossed products of Peters.
We study a class of matrix function algebras, here denoted $\mathcal{T}^{+}(\mathcal{C}_n)$. We introduce a notion of point derivations, and classify the point derivations for certain finite dimensional representations of $\mathcal{T}^{+}(\mathcal{C}_n)$. We use point derivations and information about $n \times n$ matrices to show that every $\mathcal{T}^{+}(\mathcal{C}_n)$-valued derivation on $\mathcal{T}^{+}(\mathcal{C}_n)$ is inner.
This paper focuses on certain finite dimensional point derivations for the non-selfadjoint operator algebras corresponding to directed graphs. We begin by analyzing the derivations corresponding to full matrix representations of the tensor algebra of a directed graph. We determine when such a derivation is inner, and describe situations that give rise to non-inner derivations. We also analyze the situation when the derivation corresponds to a multiplicative linear functional.
Given a directed graph, there exists a universal operator algebra and universal C*-algebra associated to the directed graph. In this paper we give intrinsic constructions of these objects. We provide an explicit construction for the maximal C*-algebra of an operator algebra. We also discuss uniqueness of the universal algebras for finite graphs, showing that for finite graphs the graph is an isomorphism invariant for the universal operator algebra of a directed graph. We show that the underlying undirected graph is a Banach algebra isomorphism invariant for the universal C*-algebra of a directed graph.
We define and investigate properties of universal operator algebras of directed graphs. Results include free products decomposition and continuity of the construction with respect to direct limits. Lastly we prove some K-theoretic results about our algebras.