Fuzzy Hilbert $C^*$-modules
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Abstract
In the present article, we introduce and study the notion of fuzzy inner product $A$-module, where $A$ is an arbitrary unital $C^*$-algebra. Moreover, we construct some examples of particular classes of $C^*$-algebras. As an application, we obtain some $M_{n}(A)$-valued fuzzy inner product, where $M_{n}(A)$ denotes the $n \times n$ matrix $C^*$-algebra of a unital $C^*$-algebra $A.$ Moreover, we obtain some relations with the notion of $C^*$-valued fuzzy normed spaces.
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References
[1] R. Ameri, Fuzzy inner product and fuzzy norm of hyperspaces, Glasnik Mathematiki. 11 (3) (2014), 125–136. https://doi.org/10.22111/IJFS.2014.1574 Google Scholar
[2] A. A. Azzam, Spherical fuzzy and soft topology: Some applications, J. Math. Comput. Sci. 32 (2) (2024), 152–159. http://dx.doi.org/10.22436/jmcs.032.02.05 Google Scholar
[3] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (3) (2003), 687–706. Google Scholar
[4] R. Chaharpashlou, D. O’Regan, C. Park, R. Saadati, C∗-Algebra valued fuzzy normed spaces with application of Hyers-Ulam stability of a random integral equation, Adv. Difference Equ. 2020 (2020), Paper No. 326, 9 pp. https://doi.org/10.1186/s13662-020-02780-0 Google Scholar
[5] A. M. El-Abyad, H. M. El-Hamouly, Fuzzy inner product spaces, Fuzzy Sets Syst. 44 (2) (1991), 309–326. https://doi.org/10.1016/0165-0114(91)90014-H Google Scholar
[6] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets Syst. 90 (3) (1997), 365–368. https://doi.org/10.1016/S0165-0114(96)00207-2 Google Scholar
[7] M. Goudarzi, S. M. Vaezpour, On the definition of fuzzy Hilbert spaces and its application, J. Nonlinear Sci. Appl. 2 (1) (2009), 46–59. http://dx.doi.org/10.22436/jnsa.002.01.07 Google Scholar
[8] M. Goudarzi, S. M. Vaezpour, R. Saadati, On the intuitionistic fuzzy inner product spaces, Chaos Solitons Fract. 41 (3) (2009), 1105–1112. https://doi.org/10.1016/j.chaos.2008.04.040 Google Scholar
[9] A. Hasankhani, A. Nazari, M. Saheli, Some properties of fuzzy Hilbert spaces and norm of operators, Iran. J. Fuzzy Syst. 7 (3) (2010), 129–157. https://doi.org/10.22111/IJFS.2010.196 Google Scholar
[10] I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (4) (1953), 839–858. Google Scholar
[11] V. A. Khan, O. Kisi, R. Akbiyik, New types of convergence of double sequences in neutrosophic fuzzy G-metric spaces, J. Nonlinear Sci. Appl. 17 (4) (2024), 150–179. http://dx.doi.org/10.22436/jnsa.017.04.01 Google Scholar
[12] C. Khaofong, K. Khammahawong, Fixed point theorems in C∗-algebra valued fuzzy metric spaces with application, Thai J. Math. 19 (3) (2021), 964–970. Google Scholar
[13] J. K. Kohli, R. Kumar, On fuzzy inner product spaces and fuzzy co-inner product spaces, Fuzzy Sets Syst. 53 (2) (1993), 227–232. https://doi.org/10.1016/0165-0114(93)90177-J Google Scholar
[14] S. G. Korma, P. R. Kishore, D. C. Kifetew, Fuzzy lattice ordered group based on fuzzy partial ordering relation, Korean J. Math. 32 (2) (2024), 195–211. https://doi.org/10.11568/kjm.2024.32.2.195 Google Scholar
[15] E. C. Lance, Hilbert C∗-Modules: A Tool for Operator Algebras, Cambridge University Press, Cambridge, 1995. Google Scholar
[16] P. Majumdar, S. K. Samanta, On fuzzy inner product spaces, J. Fuzzy Math. 16 (2008), no. 2, 377–392. Google Scholar
[17] N. Malik, M. Shabir, T. M. Al-shami, R. Gul, M. Arar, A novel decision-making technique based on T-rough bipolar fuzzy sets, J. Math. Comput. Sci. 33 (3) (2024), 275–289. http://dx.doi.org/10.22436/jmcs.033.03.06 Google Scholar
[18] E. Mostofian, M. Azhini, A. Bodaghi, Fuzzy inner product spaces and fuzzy orthogonality, Tbilisi Math. J. 10 (2) (2017), 157–171. Google Scholar
[19] S. Mukherjee, T. Bag, Fuzzy real inner product space and its properties, Ann. Fuzzy Math. Inform. 6 (2) (2013), 377–389. Google Scholar
[20] S. Mukherjee, T. Bag, Fuzzy inner product space and its properties, Int. J. Fuzzy Math. Syst. 1 (2015), 57–69. Google Scholar
[21] G. J. Murphy, C∗-Algebras and Operator Theory, Academic Press, Boston, MA, 1990. Google Scholar
[22] L. Popa, S. Lavinis, Some remarks on fuzzy Hilbert spaces, (preprint). https://doi.org/10.20944/preprints202102.0522.v1 Google Scholar
[23] N. H. M. Qumami, R. S. Jain, B. S. Reddy, On the existence for impulsive fuzzy nonlinear integro-differential equations with nonlocal conditions, J. Math. Comput. Sci. 32 (1) (2024), 13–24. http://dx.doi.org/10.22436/jmcs.032.01.02 Google Scholar
[24] A. Radharamani, A. Brindha, S. Maheswari, Fuzzy normal operator in fuzzy Hilbert space and its properties, IOSR J. Eng. 8 (7) (2018), 1–6. Google Scholar
[25] R. Saadati, S. M. Vaezpour, Some results on fuzzy Banach spaces, J. Appl. Math. Comput. 17 (1-2) (2005), 475–484. https://doi.org/10.1007/BF02936069 Google Scholar
[26] M. Saheli, A comparative study of fuzzy inner product spaces, Iran. J. Fuzzy Syst. 12 (6) (2015), 75–93. https://doi.org/10.22111/ijfs.2015.2180 Google Scholar
[27] M. P. Sunil, J. S. Kumar, On beta product of hesitancy fuzzy graphs and intuitionistic hesitancy fuzzy graphs, Korean J. Math. 31 (4) (2023), 485–494. https://doi.org/10.11568/kjm.2023.31.4.485 Google Scholar
[28] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), no. 3, 338–353. Google Scholar