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Let |${{\mathcal{A}}}$| and |${{\mathcal{B}}}$| be operator algebras with |$c_{0}$| -isomorphic diagonals and let |${{\mathcal{K}}}$| denote the compact operators. We show that if |${{\mathcal{A}}}\otimes{{\mathcal{K}}}$| and |${{\mathcal{B}}}\otimes{{\mathcal{K}}}$| are isometrically isomorphic, then |${{\mathcal{A}}}$| and |${{\mathcal{B}}}$| are isometrically isomorphic. If the algebras |${{\mathcal{A}}}$| and |${{\mathcal{B}}}$| satisfy an extra analyticity condition a similar result holds with |${{\mathcal{K}}}$| being replaced by any operator algebra containing the compact operators. For nonselfadjoint graph algebras this implies that the graph is a complete invariant for various types of isomorphisms, including stable isomorphisms, thus strengthening a recent result of Dor-On, Eilers, and Geffen. Similar results are proven for algebras whose diagonals satisfy cancellation and have |$K_{0}$| -groups isomorphic to |${{\mathbb{Z}}}$|⁠. This has implications in the study of stable isomorphisms between various semicrossed products. [ABSTRACT FROM AUTHOR]

We study two classes of operator algebras associated with a unital subsemigroup P$P$ of a discrete group G$G$: one related to universal structures and one related to co‐universal structures. First we provide connections between universal C*‐algebras that arise variously from isometric representations of P$P$ that reflect the space J$\mathcal {J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation Cλ∗(P)$\mathrm{C}^*_\lambda (P)$, and we consider the boundary quotient ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ related to the minimal boundary space. We show that ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ is co‐universal in two different classes: (a) with respect to the equivariant constructible isometric representations of P$P$; and (b) with respect to the equivariant C*‐covers of the reduced non‐selfadjoint semigroup algebra A(P)$\mathcal {A}(P)$. If P$P$ is an Ore semigroup, or if G$G$ acts topologically freely on the minimal boundary space, then ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ coincides with the usual C*‐envelope Cenv∗(A(P))$\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$ in the sense of Arveson. This covers total orders, finite type and right‐angled Artin monoids, the Thompson monoid, multiplicative semigroups of non‐zero algebraic integers, and the ax+b$ax+b$‐semigroups over integral domains that are not a field. In particular, we show that P$P$ is an Ore semigroup if and only if there exists a canonical ∗$*$‐isomorphism from ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$, or from Cenv∗(A(P))$\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$, onto Cλ∗(G)$\mathrm{C}^*_\lambda (G)$. If any of the above holds, then A(P)$\mathcal {A}(P)$ is shown to be hyperrigid. [ABSTRACT FROM AUTHOR]

In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of |$\mathrm{C}^*$| -correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C |$^*$| -correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of |$\mathrm{C}^*$| -correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana's injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid |$\mathrm{C}^*$| -correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg. [ABSTRACT FROM AUTHOR]

A higher rank numerical semigroup is a positive cone whose seminormalization is isomorphic to the free abelian semigroup. The corresponding nonselfadjoint semigroup algebras are known to provide examples that answer Arveson's Dilation Problem in the negative. Here we show that these algebras share the polydisc as the character space in a canonical way. We subsequently use this feature in order to identify higher rank numerical semigroups from the corresponding nonselfadjoint algebras. [ABSTRACT FROM AUTHOR]

Let ( G , Σ ) be a (partially) ordered abelian group with Haar measure μ , let ( A , G , α ) be a dynamical system and let A ⋊ α Σ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism A ⋊ α Σ ∼ s ( A ⊗ K ( G , Σ , μ ) ) ⋊ α ⊗ Ad ρ G , where K ( G , Σ , μ ) denotes the compact operators in the CSL algebra Alg L ( G , Σ , μ ) and ρ denotes the right regular representation of G . We also show that there exists a complete lattice isomorphism between the α ˆ -invariant ideals of A ⋊ α Σ and the ( α ⊗ Ad ρ ) -invariant ideals of A ⊗ K ( G , Σ , μ ) . Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by R . A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems ( A , G , α ) for which the identity Rad ( A ⋊ α G ) = ( Rad A ) ⋊ α G persists. A broad class of such dynamical systems is identified. [ABSTRACT FROM AUTHOR]

Using representation theory techniques we prove that various spaces of derivations or one-sided multipliers over certain operator algebras are reflexive. A sample result: any bounded local derivation (local left multiplier) on an automorphic semicrossed product C(Ω) ×σ Z+ is a derivation (resp. left multiplier). In the process we obtain various results of independent interest. In particular, we show that the finite dimensional nest representations of the tensor algebra of a topological graph separate points. [ABSTRACT FROM AUTHOR]

Abstract: Let be a unital operator algebra and let α be an automorphism of that extends to a ⁎-automorphism of its -envelope . We introduce the isometric semicrossed product and we show that . In contrast, the -envelope of the familiar contractive semicrossed product may not equal . Our main tool for calculating -envelopes for semicrossed products is a new concept, the relative semicrossed product of an operator algebra by an endomorphism. As an application of our theory, we show that if is the tensor algebra of a -correspondence and α is a ⁎-extendible automorphism of that fixes the diagonal elementwise, then the contractive semicrossed product satisfies , where denotes the Cuntz–Pimsner algebra of . This extends the main result of Davidson and Katsoulis (2010) . [Copyright &y& Elsevier]