Noncommutativity as a colimit

TL;DR

This paper demonstrates that partial algebras, including noncommutative C*-algebras, can be reconstructed as colimits of their total subalgebras, extending classical dualities and exploring functorial properties of Bohrification.

Contribution

It introduces the concept of partial C*-algebras and proves they are colimits of their total subalgebras, extending duality theories and analyzing Bohrification functoriality.

Findings

Partial Boolean algebras are colimits of their total subalgebras.

Extension of Stone and Gelfand dualities to partial algebras.

Analysis of Bohrification's functoriality on partial C*-algebras.

Abstract

Every partial algebra is the colimit of its total subalgebras. We prove this result for partial Boolean algebras (including orthomodular lattices) and the new notion of partial C*-algebras (including noncommutative C*-algebras), and variations such as partial complete Boolean algebras and partial AW*-algebras. The first two results are related by taking projections. As corollaries we find extensions of Stone duality and Gelfand duality. Finally, we investigate the extent to which the Bohrification construction, that works on partial C*-algebras, is functorial.