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Tensor products of subspace lattices and rank one density

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S. Papapanayides, I. G. Todorov

Functional Analysis · Submitted 28 Mar 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive.

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arXiv ID1203.6391 (version 3)
Pith IDCUA2GNHV, Pith's permanent record of this paper and its review
CitationI. G. Todorov, S. Papapanayides. "Tensor products of subspace lattices and rank one density." arXiv:1203.6391 (2012). https://pith.science/paper/CUA2GNHV
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,190 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.15 · mentions code or data availability 0.01.

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arXiv listing abstract v3. Results and the main claim are underlined; .

We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. As a consequence, we show that the Lattice Tensor Product Formula holds for $\Alg M$ and any other reflexive operator algebra and give several further corollaries of these results.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. result 0.98 0.95 0.68 asserted 1.00
s1 If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. result 0.76 0.94 0.09 asserted 1.00
s2 As a consequence, we show that the Lattice Tensor Product Formula holds for $\Alg M$ and any other reflexive operator algebra and give several further corollaries of these results. result 0.77 0.94 0.30 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper
Original abstract

We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. As a consequence, we show that the Lattice Tensor Product Formula holds for $\Alg M$ and any other reflexive operator algebra and give several further corollaries of these results.

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