# Quantization in \(*\)-algebras and an algebraic analog of Arveson’s extension theorem

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## Abstract

In this paper our main goal is showing that many of quantization results in functional analysis are rather algebraic. Following Esslamzadeh and Taleghani (Linear Algebra Appl 438:1372–1392, 2013), we call every subspace [resp. self-adjoint unital subspace] of a unital \(*\)-algebra, a quasi operator space [resp. quasi operator system]. Local operator systems can be realized as quasi operator spaces. Arveson’s extension theorem asserts that \(\mathcal {B}(\mathcal {H})\) is an injective object in the category of operator systems. We show that Arveson’s theorem remains valid in the much larger category of quasi operator systems. This shows that Arveson’s theorem as a non commutative extension of Hahn–Banach theorem, is of purely algebraic nature. Moreover we prove an algebraic extension of Ruan’s theorem which gives a charcterization of bounded quasi operator spaces. Then we identify the largest and the smallest of such quasi quantizations of a seminormed space \(\mathcal {X}\), which we call \(QMAX(\mathcal {X})\) and \(QMIN(\mathcal {X})\).

## Keywords

Completely positive map Quasi operator system Quasi operator space Matricially seminormed space Minimal and maximal quantization## Mathematics Subject Classification

46L07 46B40## Notes

### Acknowledgements

The authors would like to express their sincere thanks to the anonymous referee, Professors Lyudmila Turowska, Ivan G. Todorov, Kyung Hoon Han and Matin Mathieu for their useful comments. Special thanks to Professor Turowska, who patiently wrote her comments on different versions of manuscript. Section 3 of this paper was written when the first author was visiting Department of Mathematical Sciences of Chalmers University of Technology and the University of Gothenburg. The first author wishes to express his sincere thanks to the Department and Professors L. Turowska and M. Asadzadeh for their hospitality during this visit.

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