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Abstract 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. 

In this paper, we show that the semi-Dirichlet\mathrm{C}^{*}-covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet\mathrm{C}^{*}-cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product. 

Fell’s absorption principle states that the left regular representation of a group absorbs any unitary representation of the group when tensored with it. In a weakened form, this result carries over to the left regular representation of a right LCM submonoid of a group and its Nica-covariant isometric representations but it fails if the semigroup does not satisfy independence. In this paper, we explain how to extend Fell’s absorption principle to an arbitrary submonoidPof a groupGby using an enhanced version of the left regular representation. Li’s semigroup\mathrm{C}^{*}-algebra\mathrm{C}^{*}_{s}(P)and its representations appear naturally in our context. Using the enhanced left regular representation, we not only provide a very concrete presentation for the reduced object for\mathrm{C}^{*}_{s}(P)but we also resolve open problems and obtain very transparent proofs of earlier results. In particular, we address the non-selfadjoint theory and we show that the non-selfadjoint object attached to the enhanced left regular representation coincides with that of the left regular representation. We obtain a non-selfadjoint version of Fell’s absorption principle involving the tensor algebra of a semigroup and we use it to improve recent results of Clouâtre and Dor-On on the residual finite dimensionality of certain\mathrm{C}^{*}-algebras associated with such tensor algebras. As another application, we give yet another proof for the existence of a\mathrm{C}^{*}-algebra which is co-universal for equivariant, Li-covariant representations of a submonoidPof a groupG. 

Abstract We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis, Inter. Math. Res. Not. 2014 (2014), 1289–1311 relating to work of Arveson, Acta Math. 118 (1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis, J. Noncommut. Geom. 8 (2014), 771–787.