Explicit construction and uniqueness for universal operator algebras of directed graphs

The Toeplitz algebra of a Hilbert bimodule

Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras O_n, and the Cuntz-Krieger algebras O_B. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for O_\infty.

Amenability and Uniqueness for Groupoids Associated with Inverse Semigroups

We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff étale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full $C^*$-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.

Gauge Invariant Uniqueness Theorem for Corners of k-graphs

For a finitely aligned k-graph $Λ$ with X a set of vertices in $Λ$ we define a universal C*-algebra called $C^*(Λ,X)$ generated by partial isometries. We show that $C^*(Λ,X)$ is isomorphic to the corner $P_XC^*(Λ)P_X$, where $P_X$ is the sum of vertex projections in X. We then prove a version of the Gauge Invariant Uniqueness Theorem for $C^*(Λ,X)$, and then use the theorem to prove various results involving fullness, simplicity and Morita equivalence as well as new results involving symbolic dynamics.

$P$-图的规范不变唯一性定理

Gauge-invariant uniqueness theorems for $P$-graphs

本文在离散群背景下利用极大余作用证明$P$-图的规范不变唯一性定理，并给出Fell丛余泛表示的抽象刻画。

A uniqueness theorem for the Nica-Toeplitz algebra of a compactly aligned product system

Fowler introduced the notion of a product system: a collection of Hilbert bimodules $\mathbf{X}=\left\{\mathbf{X}_p:p\in P\right\}$ indexed by a semigroup $P$, endowed with a multiplication implementing isomorphisms $\mathbf{X}_p\otimes_A \mathbf{X}_q\cong \mathbf{X}_{pq}$. When $P$ is quasi-lattice ordered, Fowler showed how to associate a $C^*$-algebra $\mathcal{NT}_\mathbf{X}$ to $\mathbf{X}$, generated by a universal representation satisfying some covariance condition. In this article we prove a uniqueness theorem for these so called Nica-Toeplitz algebras.

Uniqueness, universality, and homogeneity of the noncommutative Gurarij space

We realize the noncommutative Gurarij space $\mathbb{NG}$ defined by Oikhberg as the Fra\"ıssé limit of the class of finite-dimensional $1$-exact operator spaces. As a consequence we deduce that the concommutative Gurarij space is unique up to completely isometric isomorphism, homogeneous, and universal among separable $1$-exact operator spaces. We also prove that $\mathbb{NG}$ is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal TRO to the triple envelope is an isomorphism. We deduce from this fact that $\mathbb{NG}$ does not e…

The C*-envelope of the tensor algebra of a directed graph

Given an arbitrary countable directed graph $G$ we prove the C*-envelope of the tensor algebra $T_+(G)$ coincides with the universal Cuntz-Krieger algebra associated with $G$. Our approach is concrete in nature and does not rely on Hilbert module machinery. We show how our results extend to the case of higher rank graphs, where an analogous result is obtained for the tensor algebra of a row-finite $k$-graph with no sources.

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…