Enriched topologies and topological representation of semi-unital and semi-integral quantales

Abstract This paper presents the topologization of semi-unital and semi-integral quantales by means of enriched topologies. In a first step we show that semi-unital and semi-integral quantales can be represented by a specific type of right Q 2 ˆ -algebras in Sup where Q 2 ˆ is the unitalization of the quantization of 2. In a second step we use this identification and construct the corresponding Q 2 ˆ -enriched sober spaces. On this basis semi-unital and spatial quantales are characterized by Q 2 ˆ -enriched topologies. A consequence of these constructions is a natural topologization of the quantale of all closed left (resp. right) ideals of a non-commutative and unital C ⁎ -algebra.