Quantization in $$*$$ ∗ -algebras and an algebraic analog of Arveson’s extension theorem

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 $$QMIN(\mathcal {X})$$ .