Advances in Operator Theory

A Kakutani–Mackey-like theorem

Marina Haralampidou and Konstantinos Tzironis

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We give a partial extension of a Kakutani–Mackey theorem for quasi-complemented vector spaces. This can be applied in the representation theory of certain complemented (non-normed) topological algebras. The existence of continuous linear maps, in the context of quasi-complemented vector spaces, is a very important issue in their study. Relative to this, we prove that every Hausdorff quasi-complemented locally convex space has continuous linear maps, under which a certain quasi-complemented locally convex space turns to be pre-Hilbert.

Article information

Adv. Oper. Theory, Volume 3, Number 3 (2018), 507-521.

Received: 5 December 2017
Accepted: 23 January 2018
First available in Project Euclid: 7 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46A03: General theory of locally convex spaces
Secondary: 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

(semi-)quasi-complemented vector space quasi-complementor pseudo-$H$-space automorphically perfect pair


Haralampidou, Marina; Tzironis, Konstantinos. A Kakutani–Mackey-like theorem. Adv. Oper. Theory 3 (2018), no. 3, 507--521. doi:10.15352/aot.1712-1270.

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  • Ph. Blanchard and E. Brüning, Mathematical Methods in Physics. Distributions, Hilbert Space Operators, and Variational Methods, Birkhäuser, 2003.
  • L. Drewnowski, Quasi-complements in $F$-spaces, Studia Math. 77 (1984), no. 4, 373–391.
  • M. Haralampidou, Interrelations between annihilator, dual and pseudo-H-algebras, Commun. Math. Appl. Special Issue on Algebra, Topology and Topological Algebras, Veracruz, 3 (2012), no. 1, 25–38.
  • M. Haralampidou and K. Tzironis, Representations of (non-normed) topological algebras, preprint.
  • J. Horváth, Topological Vector Spaces and Distributions, Vol. I. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966.
  • W. B. Johnson, On quasi-complements, Pasific J. Math. 48 (1973), no. 1, 113–118.
  • S. Kakutani and G. W. Mackey, Two characterizations of real Hilbert space, Ann. of Math. 45 (1944), 50–58.
  • S. Kakutani and G. W. Mackey, Ring and lattice characterizations of real Hilbert space, Amer. Math. Soc. 52 (1946), no. 8, 727–733.
  • G. Köthe, Topological Vector Spaces I, Second printing, revised, Springer-Verlag, Berlin-Heidelberg, New York 1983.
  • J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), 263–269.
  • G. W. Mackey, Isomorphisms of normed linear spaces, Ann. of Math. 43 (1942), 244–260.
  • F. J. Murray, Quasi-complements and closed projections in reflexive Banach spaces, Trans. Amer. Math. Soc. 58 (1945), 77–95.
  • S. Warner, Topological Rings, North-Holland Mathematics Studies, 178. North-Holland Publishing Co., Amsterdam, 1993.