Quantization in *-Algebras II: Archimedeanization
- PDF / 354,674 Bytes
- 9 Pages / 595.276 x 790.866 pts Page_size
- 120 Downloads / 152 Views
(0123456789().,-volV)(0123456789(). ,- volV)
RESEARCH PAPER
Quantization in *-Algebras II: Archimedeanization G. H. Esslamzadeh1 • F. Taleghani2 Received: 7 June 2020 / Accepted: 27 August 2020 Ó Shiraz University 2020
Abstract Given a matrix quasi ordered *-vector space X , we describe the Archimedeanization process of X and prove some isomorphism theorems. Then, we apply these results to construct a new proof for the algebraic analog of Arveson’s extension theorem. Also we obtain an extension of Kadison’s theorem which realizes a quasi order gauged vector space as a subspace of a commutative C -algebra. Keywords Completely positive map Quasi-operator systems Archimedeanization Mathematics Subject Classification 46L07 46B40
1 Introduction Beginning with Kadison’s characterization of commutative operator systems (Kadison 1951), ordered *-vector spaces were found a strong tool in the structure theory of operator systems. Choi and Effros constructed a non-commutative generalization of Kadison’s result by characterizing operator systems in terms of their order structure (Choi and Effros 1977). Paulsen, Todorov and Tomforde considered ordered *-vector spaces as a general framework for abstract study of operator systems and introduced various operator system structures of Archimedean ordered *-vector spaces (Paulsen and Tomforde 2009; Paulsen et al. 2011). Existing literature on ordered *-vector spaces and in particular the above mentioned references, focus around the case where the positive cone is proper, while in Esslamzadeh and Taleghani (2013, 2019), Esslamzadeh et al. (2014) the authors successfully extend several major results on operator algebras to *-algebras with not necessarily proper positive cone. In the present work which is a sequel to the
& F. Taleghani [email protected] G. H. Esslamzadeh [email protected] 1
Department of Mathematics, Faculty of Sciences, Shiraz University, Shiraz 71454, Iran
2
Department of Mathematics, Islamic Azad University, Lahijan, Iran
aforementioned papers, we investigate structure of quasi ordered *-vector spaces, that is, *-vector spaces with a not necessarily proper cone. The general question of investigating the role of algebraic structure in the fundamental results of operator theory and operator algebras is the main motivation behind the present work and Bagheri-Bardi et al. (2018, 2019), Esslamzadeh and Taleghani (2013, 2019) and Esslamzadeh et al. (2014). Some major decomposition theorems for Hilbert space operators have been extended to Baer *-rings in Bagheri-Bardi et al. (2018, 2019). Algebraic analogs of several fundamental results on operator algebras, including Choi–Effros theorem, Arveson’s extension theorem and Ruan’s theorem have been constructed in Esslamzadeh and Taleghani (2013, 2019) and Esslamzadeh et al. (2014). Here we focus on the Archimedeanization process of matrix quasi ordered *-vector spaces and its consequences. This would help us to further investigate matrix structure of quasi operator systems. We introduce our notati
Data Loading...
