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.
Similar content being viewed by others
References
N. Bourbaki, Elements of mathematics. General Topology. Part 1, Hermann, Paris; Addison–Wesley Publishing Co., Reading, Mass.–London–Don Mills, Ont., 1966.
N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 4 à 7, Lecture Notes in Mathematics, Vol. 864, Masson, Paris, 1981.
S. Dăscălescu, C. Năstăsescu and Ş. Raianu, Hopf Algebras: An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 235, Marcel Dekker, Inc., New York, 2001.
P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.
C. Heunen, Characterizations of categories of commutative C*–subalgebras, Comm. Math. Phys. 331 (2014), 215–238.
M. C. Iovanov, Co–Frobenius coalgebras, J. Algebra 303 (2006), 146–153.
I. Kaplansky, Topological rings, Amer. J. Math. 69 (1947), 153–183.
J. L. Kelley, General Topology, D. Van Nostrand Company, Inc., Toronto–New York–London, 1955.
T. Y. Lam, A First Course in Noncommutative Rings, second ed., Graduate Texts in Mathematics, Vol. 131, Springer–Verlag, New York, 2001.
S. Mac Lane, Categories for the Working Mathematician, second ed., Graduate Texts in Mathematics, Vol. 5, Springer–Verlag, New York, 1998.
D. E. Radford, Hopf Algebras, Series on Knots and Everything, Vol. 49, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
I. E. Segal, Decompositions of operator algebras. II. Multiplicity theory, Mem. Amer. Math. Soc. 9 (1951).
D. Simson, Coalgebras, comodules, pseudocompact algebras and tame comodule type, Colloq. Math. 90 (2001), 101–150.
B. Stenström, Rings of Quotients: An Introduction to Methods of Ring Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 217, Springer–Verlag, New York–Heidelberg, 1975.
S. Warner, Topological Rings, North–Holland Mathematics Studies, Vol. 178, North–Holland Publishing Co., Amsterdam, 1993.
D. Zelinsky, Rings with ideal nuclei, Duke Math. J. 18 (1951), 431–442.
D. Zelinsky, Linearly compact modules and rings, Amer. J. Math. 75 (1953), 79–90.
Author information
Authors and Affiliations
Corresponding author
Additional information
Iovanov was partially supported by the UEFISCDI [grant number PN-II-ID-PCE- 2011-3-0635], contract no. 253/5.10.2011 of CNCSIS.
Reyes was supported by NSF grant DMS-1407152.
Rights and permissions
About this article
Cite this article
Iovanov, M.C., Mesyan, Z. & Reyes, M.L. Infinite-dimensional diagonalization and semisimplicity. Isr. J. Math. 215, 801–855 (2016). https://doi.org/10.1007/s11856-016-1395-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-016-1395-5