Quantum logics with the Riesz Interpolation Property

✍ Scribed by Anatolij Dvurečenskij; Pavel Pták

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 105 KB
• Volume: 271
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310178

✦ Synopsis

Abstract

We study the quantum logics which satisfy the Riesz Interpolation Property. We call them the RIP logics. We observe that the class of RIP logics is considerable large—it contains all lattice quantum logics and, also, many (infinite) non‐lattice ones. We then find out that each RIP logic can be enlarged to an RIP logic with a preassigned centre. We continue, showing that the “nearly” Boolean RIP logics must be Boolean algebras. In a somewhat surprising contrast to this, we finally show that the attempt for the σ‐complete formulation of this result fails: We show by constructing an example that there is a non‐Boolean nearly Boolean σ‐RIP logic. As a result, there are interesting σ‐RIP logics which are intrinsically close to Boolean σ‐algebras. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)