Abstract
The present paper is devoted to the local projective positivity in the category of the local function systems or commutative local operator systems. The local projective positivity reflects the local positivity occurred in the quantizations of a projective local function system. We prove that every \(\ast\)-polynormed topology compatible with a duality results in the local projective positivity given by a filter base of the unital cones with its separated intersection. It allows to describe the local projective positivity of the local \(L^{p}\)-spaces given by a bounded or unbounded positive Radon measure on a locally compact topological space.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
M. B. Asadi, Z. Hassanpour-Yakhdani, and S. Shamloo, ‘‘A locally convex version of Kadison’s representation,’’ Positivity 24, 1449–1460 (2020).
N. Bourbaki, Elements of Mathematics, Integration, Parts I–V (Nauka, Moscow, 1967) [in Russian].
A. A. Dosiev, ‘‘Local operator spaces, unbounded operators and multinormed algebras,’’ J. Funct. Anal. 255, 1724–1760 (2008).
A. A. Dosi, ‘‘Quantum cones and their duality,’’ Houston J. Math. 39, 853–887 (2013).
A. A. Dosi, ‘‘Quantum systems and representation theorem,’’ Positivity 17, 841–861 (2013).
A. A. Dosi, ‘‘Multinormed semifinite von Neumann algebras, unbounded operators and conditional expectations,’’ J. Math. Anal. Appl. 466, 573–608 (2018).
A. A. Dosi, ‘‘Quantum system structures of quantum spaces and entanglement breaking maps,’’ Sb. Math. 210 (7), 21–93 (2019).
A. A. Dosi, ‘‘Projective positivity of the function systems,’’ Positivity 27, 47 (2023). https://doi.org/10.1007/s11117-023-00997-3
E. G. Effros and C. Webster, ‘‘Operator analogues of locally convex spaces,’’ in Operator Algebras and Applications, Vol. 495 of NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. (Springer, Berlin, 1997), pp. 163–207.
E. G. Effros and S. Winkler, ‘‘Matrix convexity: Operator analogues of the bipolar and Hahn–Banach theorems,’’ J. Funct. Anal. 144, 117–152 (1997).
S. S. Kutateladze, Fundamentals of Functional Analysis, Vol. 12 of Kluwer Texts Math. Sciences (Kluwer, Dordrecht, 1995).
P. Meyer-Neiberg, Banach Lattices (Springer, Berlin, 1991).
V. I. Paulsen and M. Tomforde, ‘‘Vector spaces with an order unit,’’ Indiana Univ. Math. J. 58, 1319–1359 (2009).
V. I. Paulsen, I. G. Todorov, and M. Tomforde, ‘‘Operator system structures on ordered spaces,’’ Proc. London Math. Soc. 102 (3), 25–49 (2011).
G. Pederson, Analysis Now, Vol. 188 of Graduate Text in Mathematics (Springer, Berlin, 1988).
H. Schaefer, Topological Vector Spaces (Springer, Berlin, 1970).
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by G. G. Amosov)
Rights and permissions
About this article
Cite this article
Dosi, A. Local Projective Positivity of Local Function Systems. Lobachevskii J Math 44, 2007–2019 (2023). https://doi.org/10.1134/S199508022306015X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S199508022306015X