Document Type: Research Paper


1 Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 ,Tehran, Iran.

2 Department of Mathematics, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad Iran.


In this paper, we introduce a generalization of Hilbert $C^*$-modules which are pre-Finsler modules, namely, $C^{*}$-semi-inner product spaces. Some properties and results of such spaces are investigated, specially the orthogonality in these spaces will be considered. We then study bounded linear operators on $C^{*}$-semi-inner product spaces.


Main Subjects

[1] M. Amyari and A. Niknam, A note on Finsler modules, Bulletin of the Iranian Mathematical Society, 29 (2003), pp. 77-81.
[2] D. Bakic and B. Guljas, On a class of module maps of Hilbert C*-modules, Mathematica communications, 7 2 (2002), pp. 177-192.
[3] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), pp. 169-172.
[4] S.S. Dragomir, Semi-Inner Products and Applications, Nova Science Publishers, Hauppauge, New York, 2004.
[5] S.S. Dragomir, J.J. Koliha, and Melbourne, Two mappings related to semi-inner products and their applications in geometry of normed linear spaces, Applications of Mathematics, 45 (2000), pp. 337-355.
[6] S.G. El-Sayyad and S.M. Khaleelulla , *-semi-inner product algebras of type(p), Zb. Rad. Prirod. Mat. Fak. Ser. Mat., 23 (1993), pp. 175-187.
[7] G.D. Faulkner, Representation of linear functionals in a Banach space, Rocky Mountain J. Math., 7 (1977), pp. 789-792.
[8] J.R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129 (1967), pp. 436-446.
[9] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61 (1947), pp. 265-292.
[10] B.E. Johnson, Centralisers and operators reduced by maximal ideals, J. London Math. Soc., 43 (1986), pp. 231-233.
[11] I. Kaplansky, Modules over operator algebras, Amer. J. Math., (75) (1953), pp. 839-858.
[12] D.O. Koehler, A note on some operator theory in certain semi-inner-product spaces, Proc. Amer. Math. Soc., 30 (1971), pp. 363-366.
[13] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc., 100 (1961), pp. 29-43.
[14] G. Lumer, On the isometries of reflexive Orlicz spaces, Ann. Inst. Fourier (Grenoble)., 13 (1963), pp. 99-109.
[15] E.C. Lance, Hilbert C*-modules . a toolkit for operator algebraists, London Math. Soc. Lecture Note Series, Cambridge Univ. Press, Cambridge, 1995.
[16] B. Nath, On generalization of semi-inner product spaces, Math. J. Okayama Univ., 15 (1971), pp. 1-6.
[17] E. Pap and R. Pavlovic, Adjoint theorem on semi-inner product spaces of type (p), Zb. Rad. Prirod. Mat. Fak. Ser. Mat., 25 (1995), pp. 39-46.
[18] W.L. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc., 182 (1973), pp. 443-468.
[19] N.C. Phillips and N. Weaver, Modules with norms which take values in a C*-algebra, Pacific J. of Maths., 185 (1998), pp. 163-181.
[20] C. Puttamadaiah and H. Gowda, On generalised adjoint abelian operators on Banach spaces, Indian J. Pure Appl. Math., 17 (1986), pp. 919-924.
[21] M.A. Rieffel, Induced representations of C*-algebras, Adv. Math., 13 (1974), pp. 176-257.
[22] B. Rzepecki, On fixed point theorems of Maia type, Publications de l'Institut Mathématique, 28 (1980), pp. 179-186.
[23] E. Torrance, Strictly convex spaces via semi-inner-product space orthogonality, Proc. Amer. Math. Soc., 26 (1970), pp. 108-110.
[24] H. Zhang and J. Zhang, Generalized semi-inner products with applications to regularized learning, J. Math. Anal. Appl., 372 (2010), pp. 181-196.