Partial isometries in an absolute order unit space
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Tusi Mathematical Research Group
ORIGINAL PAPER
Partial isometries in an absolute order unit space Anil Kumar Karn1 · Amit Kumar1 Received: 7 July 2020 / Accepted: 14 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections. Keywords Absolutely ordered space · Absolute order unit space · Absolute matrix order unit space · Order projection · Partial isometry Mathematics Subject Classification 46B40 · 46L05 · 46L30
1 Introduction Order structure is one of the basic ingredient of C∗-algebra theory. If we keep Gelfand-Neumark theorem [11] as well as Kakutani theorem [17] in one place, we can deduce that the self-adjoint part of every commutative C∗-algebra is a Banach lattice besides having some other properties. As a contrast, Kadison’s anti-lattice theorem [13] informs us that the self-adjoint part of a non-commutative C∗-algebra can not be a vector lattice. Thus the study of the order structure of a general C∗-algebra opens as an interesting area. The corresponding theory evolves a study of ordered vector spaces not having any vector lattice structure. The works of Kadison, Effros, Størmer and Pedersen, besides many others, highlight various aspects of order structure of a C∗-algebra encourage us to expect a ‘non-commutative vector lattice’ or a Communicated by Ngai-Ching Wong. * Anil Kumar Karn [email protected] Amit Kumar [email protected] 1
School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, P.O.‑Jatni, District‑Khurda, Bhubaneswar, Odisha 752050, India Vol.:(0123456789)
A. K. Karn, A. Kumar
‘near lattice’ structure in it. The monogragh [26] (and references therein), for example, is a good source of information for this purpose. Initial significant contribution in this direction begins with Kadison’s functional representation of operator ( C∗ -) algebras [14] where he proved that the self adjoint part of any operator system can be represented as the space of affine continuous real valued functions on the state space of the given operator system. (An operator system is a unital, self adjoint subspace of a unital C∗-algebra.) This seminal result turned out to be a benchmark and unfolded in the duality theory of ordered vector spaces. Early development of this theory can be found in the works of Bonsall [6, 7], Edwards [9], Ellis [10], Asimov [2, 3], Ng [25] besides many other people. However, as a breakthrough, Choi and Effros [8] characterized operator systems as matrix order unit spaces (definition is given latter). This result set another benchmark for the study of the (order theoretic) functional analysis. Besides many others, the first author started working on t
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