Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe isomorphism problem for tensor algebras of multivariable dynamical systems
176
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic 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 relating to work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis.
rate research
Read More
We prove that the isomorphism relation for separable C$^*$-algebras, and also the relations of complete and $n$-isometry for operator spaces and systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space.
We continue the study of isomorphisms of tensor algebras associated to a C*-correspondences in the sense of Muhly and Solel. Inspired by by recent work of Davidson, Ramsey and Shalit, we solve isomorphism problems for tensor algebras arising from weighted partial dynamical systems. We provide complete bounded / isometric classification results for tensor algebras arising from weighted partial systems, both in terms of the C*-correspondences associated to them, and in terms of the original dynamics. We use this to show that the isometric isomorphism and algebraic / bounded isomorphism problems are two distinct problems, that require separate criteria to be solved. Our methods yield alternative proofs to classification results for Peters semi-crossed product due to Davidson and Katsoulis and for multiplicity-free graph tensor algebras due to Katsoulis, Kribs and Solel.
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 Sehnems 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 $mathcal{N}mathcal{O}^r_X$, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C*-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C*-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where $(G, P) =(mathbb{Z},mathbb{N})$.
Let $G$ be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of $G$ on $C^*$-algebras $A$ and $B$ are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of $A$ and $B$ in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of $A$ and $B$. There is an alternative formulation of the problem: an action of the dual group $hat G$ together with a suitably equivariant unitary homomorphism of $G$ give rise to a generalized fixed-point algebra via Landstads theorem, and a problem related to the above is to produce an action of $hat G$ and two such equivariant unitary homomorphisms of $G$ that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of $A$ and $B$ is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if $G$ is discrete, this will be the case for all actions of $G$.
We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $Delta $-equivalent if they have completely isometric representations $alpha $ and $beta $ respectively and there exists a ternary ring of operators $M$ such that $alpha (A)$ (resp. $beta (B)$) is equal to the norm closure of the linear span of the set $M^*beta (B)M, $ (resp. $Malpha (A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $Delta $-equivalent, then the operator algebras $Aotimes cl K$ and $Botimes cl K$ are isomorphic. Here $cl K$ is the set of compact operators on an infinite dimensional separable Hilbert space and $otimes $ is the spatial tensor product. Conversely, if $Aotimes cl K$ and $Botimes cl K$ are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $Delta $-equivalent.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
