Injective Envelopes of Real C*- and AW*-Algebras

Abdugafur Rakhimov; Laylo Ramazonova

doi:doi:10.11648/j.ijamtp.20251101.12

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Injective Envelopes of Real C- and AW-Algebras

Abdugafur Rakhimov^*

, Laylo Ramazonova

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 1)

Received: 26 March 2025 Accepted: 8 April 2025 Published: 29 April 2025

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Abstract

Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II₁. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 1)
DOI	10.11648/j.ijamtp.20251101.12
Page(s)	19-23
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), 2025. Published by Science Publishing Group

Keywords

C*- Algebras, AW*-algebras, Injective Envelopes of Real C*-Algebras

References

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[2]	Joita M, Simon. Injective envelopes for locally C*-algebras, arXiv: 2502.08608. 2025, 1-24. https://doi.org/10.48550/arXiv.2502.08608
[3]	Hamana M. Injective envelopes of C*-algebras, Math. Soc. Japan. 1979, no. 1, 159-183.
[4]	Ayupov Sh. A, Rakhimov A. A. Real W*-algebras, Actions of groups and Index theory for real factors, VDM Publishing House Ltd. Beau-Bassin, Mauritius, 2010, Publisher Location: Publisher; Year, Page Range. 138p.
[5]	Ayupov Sh. A, Rakhimov A. A, Usmanov Sh. M. Jordan, Real and Lie Structures in Operator Algebras, Kluw. Acad. Pub., MAIA, 1997, 418, 235p.
[6]	Blecher D. P., Cecco A., Kalantar M. Real Structure in Operator Spaces, Injective Envelopes and G-spaces. Integr. Equ. Oper. Theory, 2024, 96/14. https://doi.org/10.1007/s00020-024-02766-7
[7]	Cecco A. A categorical approach to injective envelopes. Ann. Funct. Anal. 2024, 15/49.
[8]	Bryder R. S. Injective envelopes and the intersection property. https://doi.org/10.48550/arXiv.1704.02723
[9]	Paulsen V. Injective Envelopes. Cambridge University Press, Completely Bounded Maps and Operator Algebras, 2003, 206-224. https://doi.org/10.1017/CBO9780511546631.016
[10]	Rakhimov A. A, Nurillaev M. E. On property of injectivity for real W*-algebras and JW-algebras, Positivity, 2018, 22, 1345-1354. https://doi.org/10.1007/s11117-018-0579-0
[11]	Kim D. A characterization of approximately inner automorphisms of AW*-factor of type II1, Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences, 2023, 4, no. 3, 1-8.
[12]	Berberian S. K. Baer *-rings, Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag Berlin Heidelberg. 2011, 309p.
[13]	Rakhimov A. A, Ramazonova L. D. Description of the real AW*-algebras with abelian self-adjoint part, Uzbek Mathematical Journal. 2021, 65/1, 147-149.
[14]	Rakhimov A. A, Rashidova F. A. Projectionless real C*-algebras, Methods of Functional Analysis and Topology. 28/2, 2022, 144-149. https://doi.org/10.31392/MFAT-npu26-2.2022.06
[15]	Voiculescu D. A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 1976, 21, 97-113.

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Rakhimov, A., Ramazonova, L. (2025). Injective Envelopes of Real C*- and AW*-Algebras. International Journal of Applied Mathematics and Theoretical Physics, 11(1), 19-23. https://doi.org/10.11648/j.ijamtp.20251101.12

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ACS Style

Rakhimov, A.; Ramazonova, L. Injective Envelopes of Real C*- and AW*-Algebras. Int. J. Appl. Math. Theor. Phys. 2025, 11(1), 19-23. doi: 10.11648/j.ijamtp.20251101.12

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AMA Style

Rakhimov A, Ramazonova L. Injective Envelopes of Real C*- and AW*-Algebras. Int J Appl Math Theor Phys. 2025;11(1):19-23. doi: 10.11648/j.ijamtp.20251101.12

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@article{10.11648/j.ijamtp.20251101.12,
  author = {Abdugafur Rakhimov and Laylo Ramazonova},
  title = {Injective Envelopes of Real C*- and AW*-Algebras
},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {1},
  pages = {19-23},
  doi = {10.11648/j.ijamtp.20251101.12},
  url = {https://doi.org/10.11648/j.ijamtp.20251101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251101.12},
  abstract = {Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.
},
 year = {2025}
}

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TY  - JOUR
T1  - Injective Envelopes of Real C*- and AW*-Algebras

AU  - Abdugafur Rakhimov
AU  - Laylo Ramazonova
Y1  - 2025/04/29
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251101.12
DO  - 10.11648/j.ijamtp.20251101.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 19
EP  - 23
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251101.12
AB  - Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

VL  - 11
IS  - 1
ER  -

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Author Information

Abdugafur Rakhimov

Algebra and Functional Analysis, National University of Uzbekistan, Tashkent, Uzbekistan; Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey

Contact Email

http://orcid.org/0000-0002-2675-1510
Laylo Ramazonova

V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

Contact Email

http://orcid.org/0009-0009-8772-1129

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APA Style

Rakhimov, A., Ramazonova, L. (2025). Injective Envelopes of Real C*- and AW*-Algebras. International Journal of Applied Mathematics and Theoretical Physics, 11(1), 19-23. https://doi.org/10.11648/j.ijamtp.20251101.12

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ACS Style

Rakhimov, A.; Ramazonova, L. Injective Envelopes of Real C*- and AW*-Algebras. Int. J. Appl. Math. Theor. Phys. 2025, 11(1), 19-23. doi: 10.11648/j.ijamtp.20251101.12

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AMA Style

Rakhimov A, Ramazonova L. Injective Envelopes of Real C*- and AW*-Algebras. Int J Appl Math Theor Phys. 2025;11(1):19-23. doi: 10.11648/j.ijamtp.20251101.12

Copy | Download

@article{10.11648/j.ijamtp.20251101.12,
  author = {Abdugafur Rakhimov and Laylo Ramazonova},
  title = {Injective Envelopes of Real C*- and AW*-Algebras
},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {1},
  pages = {19-23},
  doi = {10.11648/j.ijamtp.20251101.12},
  url = {https://doi.org/10.11648/j.ijamtp.20251101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251101.12},
  abstract = {Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Injective Envelopes of Real C*- and AW*-Algebras

AU  - Abdugafur Rakhimov
AU  - Laylo Ramazonova
Y1  - 2025/04/29
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251101.12
DO  - 10.11648/j.ijamtp.20251101.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 19
EP  - 23
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251101.12
AB  - Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

VL  - 11
IS  - 1
ER  -

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