Korean J. Math. Vol. 27 No. 4 (2019) pp.977-1003
DOI: https://doi.org/10.11568/kjm.2019.27.4.977

Some properties of bilinear mappings on the tensor product of $C^*$-algebras

Main Article Content

Anamika Sarma
Nilakshi Goswami
Vishnu Narayan Mishra


Let $\mathcal{A}$ and $\mathcal{B}$ be two unital $C^*$-algebras and $\mathcal{A}\otimes\mathcal{B}$ be their algebraic tensor product. For two bilinear maps on $\mathcal{A}$ and $\mathcal{B}$ with some specific conditions, we derive a bilinear map on $\mathcal{A}\otimes\mathcal{B}$ and study some characteristics. Considering two $\mathcal{A}\otimes\mathcal{B}$ bimodules, a centralizer is also obtained for $\mathcal{A}\otimes\mathcal{B}$ corresponding to the given bilinear maps on $\mathcal{A}$ and $\mathcal{B}$. A relationship between orthogonal complements of subspaces of $\mathcal{A}$ and $\mathcal{B}$ and their tensor product is also deduced with suitable example.

Article Details


[1] C. A. Akemann , G. K. Pedersen and J. Tomiyama, Multipliers of C∗-algebras, J. Funct. Anal. 13 (1973), 277–301. Google Scholar

[2] A. Alahamdi, H. Alhazmi, S. Ali and A.N. Khan, A characterization of Jordan left ∗-centralizers in rings with involution, Appl. Math. Inf. Sci. 11 (2) (2017), 441–447. Google Scholar

[3] D. P. Blecher, Geometry of the tensor product of C∗-algebras, Math. Proc. Cam- bridgePhilos. Soc. (1988), 119–127. Google Scholar

[4] F. F. Bonsall and J. Duncan, Complete normed Algebras, Book, Springer- Verlag (1973). Google Scholar

[5] M. F. Bruckler , Tensor products of C∗-algebras, Operator spaces and Hilbert C∗-modules, Math. comm. 4 (1999), 257–268. Google Scholar

[6] T. K. Carne, Tensor products and Banach algebras, J. London Math. Soc. 17 (2) (1978), 480–488. Google Scholar

[7] D. Das and L. N. Mishra, Some Fixed Point Results for J HR operator pairs in C∗-algebra Valued Modular b-Metric Spaces via C∗ class functions with Applications, Advanced Studies in Contemporary Mathematics, 29 (3) (2019), pp. 383-400. Google Scholar

[8] D. Das, N. Goswami, Some fixed point theorems on the sum and product of operators in tensor product spaces, IJPAM, 10D (3), (2016), 651–663. Google Scholar

[9] D. Das, N. Goswami, V. N. Mishra, Some results on fixed point theorems in Banach algebras, International Journal of Analysis and Applications, 13 (1), (2017), 32–40. Google Scholar

[10] D. Das, N. Goswami, V. N. Mishra, Some results on the projective cone normed tensor product spaces over Banach algebras, Bol. Soc. Paran. Mat, 38 (1), (2020), 197–221. Google Scholar

[11] Deepmala, A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis, Pt. Ravishankar Shukla University, Raipur, Chhatis- garh, India., (2014), 492 010. Google Scholar

[12] Deepmala and H. K. Pathak, A study on some problems on existence of solutions for nonlinear functional-integral equations, Acta Mathematica Scientia, 33 B(5) (2013), 1305–1313. Google Scholar

[13] M. Gagne, Applications of Bilinear maps in Cryptography. UWSpace, M.sc the- sis, University of Waterloo, Ontario, Canada,(2002). Google Scholar

[14] H. Ghahramani, On centralizers of Banach algebras, The Bull. of the Malaysian Math. Soc. Series 2, 38 (1) (2015), 155–164. Google Scholar

[15] N. Goswami, A. Sarma, On a subset of invertible elements of C∗-algebras and some related fixed point theorems, Journal of Tripura Math. Soc., 19 (2017), 21–37. Google Scholar

[16] A. Guichardet , Tensor products of C∗-algebras, Arhus Universitet, Matematisk Institute, (1969). Google Scholar

[17] S. Helgason , Multipliers of Banach algebras, Ann. of Math., 64 (1956), 240–254. Google Scholar

[18] B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. Soc., 14 (1964), 299–320. Google Scholar

[19] A. S. Kavruk, Complete positivity in opeartor algebras, an M.Sc. thesis, Bilkent University, (2006). Google Scholar

[20] S. Kaijser and A. M. Sinclair, Projective tensor products of C∗-algebras, Math.Scand 65 (1984), 161–187. Google Scholar

[21] X. Liu, M. Zhou, L. N. Mishra, V. N. Mishra, B. Damjanovi c, Common fixed point theorem of six self-mappings in Menger spaces using (CLRST ) property, Open Mathematics, 2018; 16: 1423-1434. Google Scholar

[22] L. N. Mishra, M. Sen, On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order, Applied Mathematics and Computation 285 (2016), 174–183. DOI: 10.1016/j.amc.2016.03.002 Google Scholar

[23] L. N. Mishra, On existence and behavior of solutions to some nonlinear integral equations with applications, Ph.D. Thesis, National Institute of Technology, Silchar, Assam, India., (2017), 788 010. Google Scholar

[24] L. N. Mishra, K. Jyoti, A. Rani, Vandana, Fixed point theorems with digital contractions image processing, Nonlinear Sci. Lett. A, 9 (2) (2018), pp.104–115. Google Scholar

[25] L. N. Mishra, S. K. Tiwari, V. N. Mishra, I.A. Khan, Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces, Journal of Function Spaces, Volume 2015 (2015), Article ID 960827, 8 pages. Google Scholar

[26] L. N. Mishra, S. K. Tiwari, V. N. Mishra, Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces, Journal of Applied Analysis and Computation, 5 (4) (2015), pp. 600–612. doi:10.11948/2015047 Google Scholar

[27] N. M. McCullagh, Cryptographic Applications of Bilinear Maps, Ph.D. Thesis, Dublin City University, Dublin, Ireland (2005). Google Scholar

[28] H. K. Pathak and Deepmala, Common fixed point theorems for PD-operator pairs under Relaxed conditions with applications, Journal of Computational and Applied Mathematics, 239 (2013), 103–113. Google Scholar

[29] A. R. Raymond, Introduction to Tensor products of Banach Spaces, Book Springer - Verlag London Limited, (2001). Google Scholar

[30] A. Sarma, N. Goswami, Some results on positive elements in the tensor product of C∗-algebras, Int. J. Math. Trends and Technology, 51 (3), (2017), 237–243. Google Scholar

[31] M. S. T. EI-Sayiad, N. M. Muthana, Z. S. Alkhamisi, A note on Jordan left ∗-centralizers on prime and semiprime rings with involution, J. Taibah Univ.Sci., (2018), 1080–1082. Google Scholar

[32] R. Schwartz, Notes on Tensor products, (2014), https://www.math.brown.edu/ res/M153/tensor.pdf Google Scholar

[33] G. N. Prasanth, Introduction to bilinear forms, Scholarly project, Govt. College of Chittur, (2014) https://www.researchgate.net/publication/267982491. Google Scholar

[34] T. Turumaru, On the direct product of operator algebras I, Tohoku Math. Journ., Google Scholar

[35] (1952), 242–151. Google Scholar