Topologized Hilbert spaces

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Topologized Hilbert spaces

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作者：Chi-Keung Ng ^{ckng@nankai.edu.cn" class="auth_mail}
关键词：Hilbert spaces ; Densely defined operators ; *-Algebras ; Banach *-algebras
刊名：Journal of Mathematical Analysis and Applications
出版年：1 October 2014
年：2014
卷：418
期：1
页码：108-120
全文大小：376 K

文摘

Let X14003011&_mathId=si1.gif&_user=111111111&_pii=S0022247X14003011&_rdoc=1&_issn=0022247X&md5=6a29bfd2b7d35efd57537c69de38f0ae" title="Click to view the MathML source">H_P

<

mi mathvariant="script">Hmi><mi mathvariant="script">Pmi> be a Hausdorff topological vector space with the underlying vector space H

<

mi mathvariant="script">Hmi> being a Hilbert space such that P

<

mi mathvariant="script">Pmi> is coarser than the norm topology. A densely defined P

<

mi mathvariant="script">Pmi>-P

<

mi mathvariant="script">Pmi>-continuous operator on H

<

mi mathvariant="script">Hmi> is called P

<

mi mathvariant="script">Pmi>-maximal if it has no non-trivial P

<

mi mathvariant="script">Pmi>-P

<

mi mathvariant="script">Pmi>-continuous extension, and it is said to be P

<

mi mathvariant="script">Pmi>-adjointable if its adjoint is also P

<

mi mathvariant="script">Pmi>-P

<

mi mathvariant="script">Pmi>-continuous.

We show that if P $<$ mi mathvariant="script">Pmi> is locally convex, the collection $<$ mi mathvariant="fraktur">Mmi><mi mathvariant="script">Pmi>鈰?/mo>(<mi mathvariant="script">Hmi>) of all densely defined P $<$ mi mathvariant="script">Pmi>-maximal P $<$ mi mathvariant="script">Pmi>-adjointable operators is a ^鈦?/sup> $^{鈦?/mo>}$ -algebra under the multiplication given by the P $<$ mi mathvariant="script">Pmi>-maximal extension of the composition and the involution ^鈰?/sup> $^{鈰?/mo>}$ given by the P $<$ mi mathvariant="script">Pmi>-maximal extension of the adjoint. Examples include rigged Hilbert spaces and O^鈦?/sup> $<$ mi>Omi>鈦?/mo>-algebras.

In the general (not necessarily locally convex) case, we associate with H_P $<$ mi mathvariant="script">Hmi><mi mathvariant="script">Pmi> a ^鈦?/sup> $^{鈦?/mo>}$ -algebra $<$ mi mathvariant="fraktur">Lmi><mi mathvariant="bold">bmi>鈰?/mo>(<mi mathvariant="script">Hmi><mi mathvariant="script">Pmi> ˜) which is a ^鈦?/sup> $^{鈦?/mo>}$ -subalgebra of $<$ mi mathvariant="fraktur">Mmi><mi mathvariant="script">Pmi>鈰?/mo>(<mi mathvariant="script">Hmi>) when P $<$ mi mathvariant="script">Pmi> is locally convex. If P $<$ mi mathvariant="script">Pmi> is the measure topology on H $<$ mi mathvariant="script">Hmi> corresponding to a tracial von Neumann algebra M⊆L(H) $<$ mi mathvariant="script">Mmi>⊆<mi mathvariant="fraktur">Lmi>(<mi mathvariant="script">Hmi>), then the image of the representation of the measurable operator algebra on the completion $<$ mi mathvariant="script">Hmi><mi mathvariant="script">Pmi> ˜ of H $<$ mi mathvariant="script">Hmi> with respect to P $<$ mi mathvariant="script">Pmi>, can be regarded as a ^鈦?/sup> $^{鈦?/mo>}$ -subalgebra of $<$ mi mathvariant="fraktur">Lmi><mi mathvariant="bold">bmi>鈰?/mo>(<mi mathvariant="script">Hmi><mi mathvariant="script">Pmi> ˜).

In the case when P $<$ mi mathvariant="script">Pmi> is normable, it is shown that $<$ mi mathvariant="fraktur">Lmi><mi mathvariant="bold">bmi>鈰?/mo>(<mi mathvariant="script">Hmi><mi mathvariant="script">Pmi> ˜) is a Banach ^鈦?/sup> $^{鈦?/mo>}$ -algebra. Examples of such Banach ^鈦?/sup> $^{鈦?/mo>}$ -algebras include $<$ mi mathvariant="fraktur">Lmi><mi>Lmi>∞[0,1]鈰?/mo>(<mi>Lmi>2[0,1]):={<mi>唯mi>∈<mi mathvariant="fraktur">Bmi>(<mi>Lmi>2[0,1]):<mi>唯mi>(<mi>Lmi>∞[0,1])⊆<mi>Lmi>∞[0,1];<mi>唯mi>鈦?/mo>(<mi>Lmi>∞[0,1])⊆<mi>Lmi>∞[0,1]} (under a suitable norm) as well as $<$ mi mathvariant="fraktur">Lmi><mi mathvariant="fraktur">Tmi>(<mi>鈩?/mi>2)鈰?/mo>(<mi mathvariant="fraktur">Smi>(<mi>鈩?/mi>2)):={<mi>桅mi>∈<mi mathvariant="fraktur">Bmi>(<mi mathvariant="fraktur">Smi>(<mi>鈩?/mi>2)):<mi>桅mi>(<mi mathvariant="fraktur">Tmi>(<mi>鈩?/mi>2))⊆<mi mathvariant="fraktur">Tmi>(<mi>鈩?/mi>2);<mi>桅mi>鈦?/mo>(<mi mathvariant="fraktur">Tmi>(<mi>鈩?/mi>2))⊆<mi mathvariant="fraktur">Tmi>(<mi>鈩?/mi>2)}, where S(鈩?sup>2) $<$ mi mathvariant="fraktur">Smi>(<mi>鈩?/mi>2) and 14208768a1ce56f588" title="Click to view the MathML source">T(鈩?sup>2) $<$ mi mathvariant="fraktur">Tmi>(<mi>鈩?/mi>2) are the spaces of Hilbert–Schmidt operators and of trace-class operators respectively, on 鈩?sup>2 $<$ mi>鈩?/mi>2.

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