On unitary representability of topological groups |
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Authors: | Jorge Galindo |
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Institution: | (1) Departmento de Matemáticas, Universidad Jaume I, Campus Riu Sec, 12071 Castellón, Spain |
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Abstract: | We prove that the additive group (E*, τ
k
(E)) of an -Banach space E, with the topology τ
k
(E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ
k
(E)) is uniformly homeomorphic to a subset of for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability
of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above
facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.
Research partially supported by Spanish Ministry of Science, grant MTM2008-04599/MTM. The foundations of this paper were laid
during the author’s stay at the University of Ottawa supported by a Generalitat Valenciana grant CTESPP/2004/086. |
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Keywords: | Unitary group Positive definite -Banach space" target="_blank">gif" alt="$${\mathcal {L}_\infty}$$" align="middle" border="0">-Banach space Free Abelian topological group Free locally convex space Free Banach space Unitarily representable Uniform embedding Schwartz space |
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