scholarly journals The isomorphism problem for some universal operator algebras

2011 ◽  
Vol 228 (1) ◽  
pp. 167-218 ◽  
Author(s):  
Kenneth R. Davidson ◽  
Christopher Ramsey ◽  
Orr Moshe Shalit
Keyword(s):  
Operator Algebras ◽  
Isomorphism Problem ◽  
Universal Operator

COMPLETE CLASSIFICATION OF SPECTRAL OPERATOR ALGEBRAS ASSOCIATED WITH DIVISIBLE PRODUCT SYSTEMS

1998 ◽  
Vol 09 (08) ◽  
pp. 923-943
Author(s):  
MASAYASU AOTANI
Keyword(s):  
Operator Algebras ◽  
Banach Algebras ◽  
Complete Classification ◽  
Isomorphism Problem ◽  
Spectral Operator ◽  
Product System ◽  
C Algebra ◽  
Product Systems ◽  
Algebraic Representations

Completely spatial E0-semigroups constitute the most important class of E0-semigroups. Each completely spatial E0-semigroup α induces a divisible product system Eα and a C*-algebra C*(Eα) called the spectral C*-algebra. It has been shown by Arveson that Eα and Eβ are isomorphic as product systems if and only if α and β are cocycle conjugate. He has also proved that representations of E correspond bijectively to ordinary C*-algebraic representations of C*(E). While it is trivial to show that C*(Eα) and C*(Eβ) are isomorphic if the underlying product systems Eα and Eβ are isomorphic, it is not known whether C*(Eα) and C*(Eβ) can be isomorphic when Eα and Eβ are not. In this paper we will consider a related isomorphism problem among the Banach algebras, known as spectral operator algebras, associated with divisible product systems. It will be shown that the spectral operator algebras [Formula: see text] and [Formula: see text] are isomorphic if and only if Eα and Eβ are isomorphic. This classification is important as C*(E) is a hereditary subalgebra of the C*-algebra [Formula: see text] generated by [Formula: see text].

scholarly journals The Non-selfadjoint Approach to the Hao–Ng Isomorphism

2019 ◽  
Author(s):  
Elias G Katsoulis ◽  
Christopher Ramsey
Keyword(s):  
Discrete Group ◽  
Operator Algebras ◽  
Finite Graph ◽  
Isomorphism Problem ◽  
Crossed Product ◽  
Compact Groups ◽  
Crossed Products ◽  
Injective Envelope ◽  
Tensor Algebras ◽  
Dilation Theory

Abstract 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.

scholarly journals The isomorphism problem for k-trees is complete for logspace

2012 ◽  
Vol 217 ◽  
pp. 1-11 ◽  
Author(s):  
V. Arvind ◽  
Bireswar Das ◽  
Johannes Köbler ◽  
Sebastian Kuhnert
Keyword(s):  
Isomorphism Problem
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