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This article is cited in 14 scientific papers (total in 14 papers)
Metric freeness and projectivity for classical and quantum normed modules
A. Ya. Helemskii M. V. Lomonosov Moscow State University
Abstract:
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 $l_1(M)$ (where $M$ is a set) that are denoted by $l_1^0(M)$ and consist of finitely supported functions. Thus, in this case, projectivity coincides with freeness.
Bibliography: 28 titles.
Keywords:
quantum module, metric projectivity, freeness, rigging, asymptotic structure.
Received: 09.01.2012 and 12.12.2012
Citation:
A. Ya. Helemskii, “Metric freeness and projectivity for classical and quantum normed modules”, Mat. Sb., 204:7 (2013), 127–158; Sb. Math., 204:7 (2013), 1056–1083
Linking options:
https://www.mathnet.ru/eng/sm8101https://doi.org/10.1070/SM2013v204n07ABEH004330 https://www.mathnet.ru/eng/sm/v204/i7/p127
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Abstract page: | 556 | Russian version PDF: | 235 | English version PDF: | 7 | References: | 48 | First page: | 46 |
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