Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full <I>C</I><SUP>*</SUP>-Algebras

Weijiao He

doi:doi:10.11648/j.ijtam.20210702.11

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Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C^*-Algebras

Weijiao He

Published in International Journal of Theoretical and Applied Mathematics (Volume 7, Issue 2)

Received: 11 January 2021 Accepted: 18 March 2021 Published: 7 April 2021

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Abstract

In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring’s Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 7, Issue 2)
DOI	10.11648/j.ijtam.20210702.11
Page(s)	17-29
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Fell Bundles, Schur Multipliers, Herz-Schur Multipliers, Nuclearity of C^*-algebras, Approximation Property

References

[1]	Marek Boz•ejko and Gero Fendler, Herz-schur multipliers and completely bounded multipliers of the fourier algebra of a locally compact group, Unione Matematica Italiana. Bollettino. A. Serie VI 3 (1984), no. 2 (1984).
[2]	Nathanial P. Brown and Narutaka Ozawa, C^*-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR2391387.
[3]	Jean De Canniere and Uffe Haagerup, Multipliers of the fourier algebras of some simple lie groups and their discrete subgroups, American Journal of Mathematics 107 (1985).
[4]	Ruy Exel, Partial dynamical systems, Fell bundles and applications, Mathematical Surveys and Monographs, vol. 224, American Mathematical Society, Providence, RI, 2017. MR3699795.
[5]	Ruy Exel and Chi-Keung Ng, Approximation property of C^*-algebraic bundles, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 3, 509-522. MR1891686.
[6]	J. M. G. Fell and R. S. Doran, Representations of ∗-algebras, locally compact groups, and Banach C^*- algebraic bundles. Vol. 1, Pure and Applied Mathematics, vol. 125, Academic Press, Inc., Boston, MA, 1988. Basic representation theory of groups and algebras. MR936628.
[7]	J. M. G. Representations of ^∗-algebras, locally compact groups, and Banach ^∗-algebraicbundles. Vol. 2, Pureand Applied Mathematics, vol. 126, Academic Press, Inc., Boston, MA, 1988. Banach ^∗-algebraic bundles, induced representations, and the generalized Mackey analysis. MR936629.
[8]	A. Grothendieck, Rsum de la thorie mtrique des produits tensoriels topologiques, Boll. Soc. Mat. Sao-Paulo 8 (1956), 1-79.
[9]	Andrew McKee, Adam Skalski, Ivan G. Todorov, and Lyudmila Turowska, Positive Herz-Schur multipliers and approximation properties of crossed products, Math. Proc. Cambridge Philos. Soc. 165 (2018), no. 3, 511- 532. MR3860401.
[10]	Andrew McKee, Ivan Todorov, and Lyudmyla Turowska, Herz-Schur multipliers of dynamical systems (2016). Preprint. arXiv:1608.01092 [math.OA].
[11]	Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Ad-vanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR1976867.

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Weijiao He. (2021). Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras. International Journal of Theoretical and Applied Mathematics, 7(2), 17-29. https://doi.org/10.11648/j.ijtam.20210702.11

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Weijiao He. Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras. Int. J. Theor. Appl. Math. 2021, 7(2), 17-29. doi: 10.11648/j.ijtam.20210702.11

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AMA Style

Weijiao He. Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras. Int J Theor Appl Math. 2021;7(2):17-29. doi: 10.11648/j.ijtam.20210702.11

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@article{10.11648/j.ijtam.20210702.11,
  author = {Weijiao He},
  title = {Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {7},
  number = {2},
  pages = {17-29},
  doi = {10.11648/j.ijtam.20210702.11},
  url = {https://doi.org/10.11648/j.ijtam.20210702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20210702.11},
  abstract = {In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring’s Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.},
 year = {2021}
}

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TY  - JOUR
T1  - Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras
AU  - Weijiao He
Y1  - 2021/04/07
PY  - 2021
N1  - https://doi.org/10.11648/j.ijtam.20210702.11
DO  - 10.11648/j.ijtam.20210702.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
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PB  - Science Publishing Group
SN  - 2575-5080
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AB  - In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring’s Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.
VL  - 7
IS  - 2
ER  -

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Author Information

Weijiao He

The Department of Mathematics, Taiyuan Normal University, Taiyuan, China

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Weijiao He. (2021). Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras. International Journal of Theoretical and Applied Mathematics, 7(2), 17-29. https://doi.org/10.11648/j.ijtam.20210702.11

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ACS Style

Weijiao He. Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras. Int. J. Theor. Appl. Math. 2021, 7(2), 17-29. doi: 10.11648/j.ijtam.20210702.11

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AMA Style

Weijiao He. Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras. Int J Theor Appl Math. 2021;7(2):17-29. doi: 10.11648/j.ijtam.20210702.11

Copy | Download

@article{10.11648/j.ijtam.20210702.11,
  author = {Weijiao He},
  title = {Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {7},
  number = {2},
  pages = {17-29},
  doi = {10.11648/j.ijtam.20210702.11},
  url = {https://doi.org/10.11648/j.ijtam.20210702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20210702.11},
  abstract = {In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring’s Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras
AU  - Weijiao He
Y1  - 2021/04/07
PY  - 2021
N1  - https://doi.org/10.11648/j.ijtam.20210702.11
DO  - 10.11648/j.ijtam.20210702.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 17
EP  - 29
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20210702.11
AB  - In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring’s Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.
VL  - 7
IS  - 2
ER  -

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