Journal article
Infinite-dimensional diagonalization and semisimplicity
Israel journal of mathematics, Vol.215(2), pp.801-855
09/01/2016
DOI: 10.1007/s11856-016-1395-5
Abstract
We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the "finite topology") on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn-Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.
Details
- Title: Subtitle
- Infinite-dimensional diagonalization and semisimplicity
- Creators
- Miodrag C Iovanov - University of IowaZachary Mesyan - University of Colorado BoulderManuel L Reyes - Bowdoin College
- Resource Type
- Journal article
- Publication Details
- Israel journal of mathematics, Vol.215(2), pp.801-855
- DOI
- 10.1007/s11856-016-1395-5
- ISSN
- 0021-2172
- eISSN
- 1565-8511
- Publisher
- HEBREW UNIV MAGNES PRESS
- Number of pages
- 55
- Grant note
- 253/5.10.2011 / CNCSIS; Consiliul National al Cercetarii Stiintifice (CNCS) DMS-1407152 / NSF; National Science Foundation (NSF) 1407152 / Division Of Mathematical Sciences; National Science Foundation (NSF); NSF - Directorate for Mathematical & Physical Sciences (MPS) PN-II-ID-PCE-2011-3-0635 / UEFISCDI; Consiliul National al Cercetarii Stiintifice (CNCS); Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii (UEFISCDI)
- Language
- English
- Date published
- 09/01/2016
- Academic Unit
- Mathematics
- Record Identifier
- 9984240765402771
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