TY - JOUR
T1 - Hilbert bundles with ends
AU - Kato, Tsuyoshi
AU - Kishimoto, Daisuke
AU - Tsutaya, Mitsunobu
N1 - Publisher Copyright:
© 2024 World Scientific Publishing Company.
PY - 2024/6/1
Y1 - 2024/6/1
N2 - Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.
AB - Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.
KW - Fourier transform
KW - Hilbert bundle
KW - end
KW - harmonic oscillator
KW - pushforward of a vector bundle
KW - spectral decomposition
KW - uniform Roe algebra
KW - unitary operator of finite propagation
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U2 - 10.1142/S1793525321500680
DO - 10.1142/S1793525321500680
M3 - Article
AN - SCOPUS:85122324297
SN - 1793-5253
VL - 16
SP - 291
EP - 322
JO - Journal of Topology and Analysis
JF - Journal of Topology and Analysis
IS - 2
ER -