Semigroupoid C*-Algebras

Interpolation in Semigroupoid Algebras

A seminal result of Agler characterizes the so-called Schur-Agler class of functions on the polydisk in terms of a unitary colligation transfer function representation. We generalize this to the unit ball of the algebra of multipliers for a family of test functions over a broad class of semigroupoids. There is then an associated interpolation theorem. Besides leading to solutions of the familiar Nevanlinna-Pick and Caratheodory-Fejer interpolation problems and their multivariable commutative and noncommutative generalizations, this approach also covers more exotic examples.

Operator algebras associated with graphs and categories of paths: a Survey

Many interesting examples of operator algebras, both self-adjoint and non-self-adjoint, can be constructed from directed graphs. In this survey, we overview the construction of $C^*$-algebras from directed graphs and from two generalizations of graphs: higher rank graphs and categories of paths. We also look at free semigroupoid algebras generated from graphs and higher rank graphs, with an emphasis on the left regular free semigroupoid algebra. We give examples of specific graphs and the algebras they generate, and we discuss properties such as semisimplicity and reflexivity. Finally, we prop…

Partly Free Algebras from Directed Graphs

We say that a nonselfadjoint operator algebra is partly free if it contains a free semigroup algebra. Motivation for such algebras occurs in the setting of what we call free semigroupoid algebras. These are the weak operator topology closed algebras generated by the left regular representations of semigroupoids associated with finite or countable directed graphs. We expand our analysis of partly free algebras from previous work and obtain a graph-theoretic characterization of when a free semigroupoid algebra with countable graph is partly free. This analysis carries over to norm closed quiver …

Isomorphisms of algebras associated with directed graphs

In this paper we verify that the graph forms a complete invariant for Banach algebra isomorphisms of tensor algebras of graphs. For tensor algebras associated with countable directed graphs having no sinks the graph forms an invariant for algebraic isomorphisms as well. The graph also forms a complete invariant for w*-bicontinuous isomorphisms of free semigroupoid algebras. For free semigroupoid algebras associated with countable directed locally finite graphs with no sinks, the graph also forms an invariant for algebraic isomorphisms.

自由半群胚代数与第一上同调群

Free semigroupoid algebras and the first cohomology groups

本文研究有向图对应的自由半群胚代数与张量代数的导子，证明强连通情形的弱Dixmier逼近定理，刻画有限图第一上同调群平凡的充要条件，引入交错数探讨无限图情形，并给出非平凡上同调的例子。

A Continuous Field of C*-algebras and the Tangent Groupoid for Manifolds with Boundary

For a smooth manifold with boundary we construct a semigroupoid and a continuous field of C*-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of \hbar-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.

Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of op…

Commutants of weighted shift directed graph operator algebras

We consider non-selfadjoint operator algebras $\mathfrak{L}(G,λ)$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for $\mathfrak{L}(G,λ)$ in the case of the single vertex graph with …

The $H^\infty$ algebras of higher rank graphs

We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hilbert space and creation operators which are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a 'higher rank semigroupoid algebra'. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs …

Nest representations of directed graph algebras

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra $\flgee$ of countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra $\T^{+}(G)$. We prove that the finite dimensional nest representations separate the points in $\flgee$, and a fortiori, in $\T^{+}(G)$. The irreducible finite dimensional representations separate the points in $\flgee$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every verte…

Inclusions of second quantization algebras

In this note we study inclusions of second quantization algebras, namely inclusions of von Neumann algebras on the Fock space of a separable complex Hilbert space H, generated by the Weyl unitaries with test functions in closed, real linear subspaces of H. We show that the class of irreducible inclusions of standard second quantization algebras is non empty, and that they are depth two inclusions, namely the third relative commutant of the Jones' tower is a factor. When the smaller vector space has codimension n into the bigger, we prove that the corresponding inclusion of second quantization …

On the classification of simple approximately subhomogeneous C*-algebras not necessarily of real rank zero

A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely the class of simple C*-algebras which are inductive limits of continuous-trace C*-algebras whose building blocks have their spectrum homeomorphic to the interval [0,1] or to a finite disjoint union of closed intervals. In particular, a classification of those stably AI algebras which are inductive limits of hereditary sub-C*-algebras of interval algebras is obtained. Also, the range of the invariant is calculated.

Decomposition rank of approximately subhomogeneous C*-algebras

It is shown that every Jiang-Su stable approximately subhomogeneous C*-algebra has finite decomposition rank. Previously, it was not even known that such algebras have finite nuclear dimension. A key step in the proof is that subhomogeneous C*-algebra are locally approximated by a certain class of more tractable subhomogeneous algebras, namely, a non-commutative generalization of the class of cell complexes. The result is applied to show that Jiang-Su stable minimal Z-crossed products have finite decomposition rank.

On the K-theory of graph C*-algebras

We classify graph C*-algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph. This is done by a purely graph theoretical calculation of the K-theory and the position of the unit therein.

Spectral theorem approach to commutative C* -algebras generated by Toeplitz operators on the unit ball: quasi-elliptic related cases

We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.