Metric freeness and projectivity for classical and quantum normed modules

Journal Article · Wed Jul 31 04:00:00 EDT 2013 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM2013V204N07ABEH004330· OSTI ID:22317923

Helemskii, A Ya ^[1]

---

1. M. V. Lomonosov Moscow State University, Moscow (Russian Federation)

In functional analysis, there are several diverse approaches to the notion of projective module. We show that a certain general categorical scheme contains all basic versions as special cases. In this scheme, the notion of free object comes to the foreground, and, in the best categories, projective objects are precisely retracts of free ones. We are especially interested in the so-called metric version of projectivity and characterize the metrically free classical and quantum (= operator) normed modules. Informally speaking, so-called extremal projectivity, which was known earlier, is interpreted as a kind of 'asymptotical metric projectivity'. In addition, we answer the following specific question in the geometry of normed spaces: what is the structure of metrically projective modules in the simplest case of normed spaces? We prove that metrically projective normed spaces are precisely the subspaces of l{sub 1}(M) (where M is a set) that are denoted by l{sub 1}{sup 0}(M) and consist of finitely supported functions. Thus, in this case, projectivity coincides with freeness. Bibliography: 28 titles.

• OSTI ID: 22317923
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 7 Vol. 204; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
BIBLIOGRAPHIES
FUNCTIONAL ANALYSIS
FUNCTIONS
GEOMETRY
MATHEMATICAL SPACE
METRICS
QUANTUM OPERATORS