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

Tensor algebras of product systems and their C*-envelopes

Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned product system over $P$. We show that the C*-envelope of the Nica tensor algebra $\mathcal{N}\mathcal{T}^+_X$ coincides with both Sehnem's covariance algebra $\mathcal{A} \times_X P$ and the co-universal C*-algebra $\mathcal{N}\mathcal{O}^r_X$ for injective, gauge compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of $\mathcal{N}\math…

图上自相似半群作用的 Wold 分解与 C*-包络

Wold Decomposition and C*-envelopes of Self-Similar Semigroup Actions on Graphs

本文建立图上自相似半群作用表示的 Wold 分解，并利用显式非平凡膨胀证明 $\mathcal{A}_{\mathbb{N},E}$ 的 C*-包络就是 $\mathcal{O}_{\mathbb{Z},E}$。

C*-envelopes of tensor algebras of product systems

Let $P$ be a submonoid of a group $G$ and let $\mathcal{E}=(\mathcal{E}_p)_{p\in P}$ be a product system over $P$ with coefficient C*-algebra $A$. We show that the following C*-algebras are canonically isomorphic: the C*-envelope of the tensor algebra $\mathcal{T}_λ(\mathcal{E})^+$ of $\mathcal{E}$; the reduced cross sectional C*-algebra of the Fell bundle associated to the canonical coaction of $G$ on the covariance algebra $A\times_{\mathcal{E}}P$ of $\mathcal{E}$; and the C*-envelope of the cosystem obtained by restricting the canonical gauge coaction on $\mathcal{T}_λ(\mathcal{E})$ to the …

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.

Embeddings and $C^*$-envelopes of exact operator systems

We prove a necessary and sufficient condition for embeddability of an operator system into $\mathcal{O}_2$. Using Kirchberg's theorems on a tensor product of $\mathcal{O}_2$ and $\mathcal{O}_{\infty}$, we establish results on their operator system counterparts $\mathcal{S}_2$ and $\mathcal{S}_{\infty}$. Applications of the results proved, including some examples describing $C^*$-envelopes of operator systems, are also discussed.

C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittade…

Finite dimensional point derivations for graph algebras

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.

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…

Full Cuntz-Krieger dilations via non-commutative boundaries

We apply Arveson's non-commutative boundary theory to dilate every Toeplitz-Cuntz-Krieger family of a directed graph $G$ to a full Cuntz-Krieger family for $G$. We do this by describing all representations of the Toeplitz algebra $\mathcal{T}(G)$ that have unique extension when restricted to the tensor algebra $\mathcal{T}_+(G)$. This yields an alternative proof to a result of Katsoulis and Kribs that the $C^*$-envelope of $\mathcal T_+(G)$ is the Cuntz-Krieger algebra $\mathcal O(G)$. We then generalize our dilation results further, to the context of colored directed graphs, by investigating …

Operator algebras and representations from commuting semigroup actions

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 cros…

Dilation theory, commutant lifting and semicrossed products

We take a new look at dilation theory for nonself-adjoint operator algebras. Among the extremal (co)extensions of a representation, there is a special property of being fully extremal. This allows a refinement of some of the classical notions which are important when one moves away from standard examples. We show that many algebras including graph algebras and tensor algebras of C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory. This leads to variations of the notions of commutant lifting and Ando's theorem. Th…

具有量子对称性的代数与范畴的内射性

Injectivity for algebras and categories with quantum symmetry

本文证明了含幺Yetter-Drinfeld C*-代数及相关双模范畴的内射包存在性，并由此得出紧量子群Drinfeld double边界作用的幺单不变性。

剩余有限维算子代数的C*-代数的余泛性

Couniversality for C*-algebras of residually finite-dimensional operator algebras

本文研究了附着于固定算子代数的剩余有限维（RFD）C*-代数空间中的余泛存在性，构造了不存在最小RFD C*-代数的实例，并证明了在大量张量代数和圆盘代数情形下RFD C*-代数空间在C*-覆盖下确界下不封闭。