Representations of pseudo Vitali spaces and Loomis–Sikorski theorem

Abstract

We present pseudo Vitali spaces as lattices with a partially defined operation, +, that is not necessarily commutative. We show that they can be embedded into ℓ-groups, and we give a categorical equivalence of the category of pseudo Vitali spaces. Studying states on Vitali spaces with strong unit, we show that every Dedekind σ-complete Vitali space is an epimorphic image of a Vitali space of bounded functions where the operations on functions are defined pointwise.

Keywords:

• pseudo Vitali space
• state
• ℓ-group
• Loomis–Sikorski theorem
• g-Vitali space

Acknowledgements

This paper has been supported by the Centre of Excellence SAS, the grant VEGA No 2/0032/09 SAV, the Slovak Research and Development Agency under contract No. APVV-0071-06, MURST, project ‘Analisi Reale’.

Notes

1. [email protected]

Additional information

Notes on contributors

F. Ventriglia

1 1. [email protected]