Document Type : Research Paper


Department of Mathematics, University of Birjand, Birjand, P. O. Box 414, 9717851367, Birjand, Iran.


This article introduces the notion of L$_p$-C$^*$-semi-inner product space, a generalization of the concept of C$^*$-semi-inner product space  introduced by Gamchi et al., where we consider H\"{o}lder's inequality instead of Cauchy Schwartz' inequality. We establish some basic results L$_p$-C$^*$-semi-inner product spaces, analogous to those valid for C$^*$-semi-inner product spaces and Hilbert C$^*$-modules.


[1] M.A. Abo-Hadi, Semi-inner product spaces of type $(p)$, l M. Sc. Diss. thesis, Faculty of Science, King Abdulaziz University, 1983.
[2] D.A. Abulhamil, F.B. Jamjoom and A.M. Peralta, Linear maps which are anti-derivable at zero, Bull. Malays. Math. Sci. Soc., 43.6 (2020), pp. 4315-4334.
[3] S. Ayupov and K. Kudaybergenov, $2-$local derivations on von Neumann algebras. Positivity, 19.3 (2015), pp. 445-455.
[4] S. Ayupov, K. Kudaybergenov and A.M. Peralta, A survey on local and $2-$local derivations on C$^*-$ and von Neumann algebras, Topics in Functional Analysis and Algebra, Contemp. Math., 672 (2016), pp. 73-126.
[5] M. Bresar, Jordan derivations on semiprime rings, Proc. Am. Math. Soc., 104.4 (1988), pp. 1003-1006.
[6] B. Fadaee and H. Ghahramani, Linear maps on C$^*$-algebras behaving like (anti-)derivations at orthogonal elements, Bull. Malays. Math. Sci. Soc., 43.3 (2020), pp. 2851-2859.
[7] S.S. Gamchi, M. Janfada and A. Niknam, C$^*$-semi-inner product spaces, Sahand Commun. Math. Anal., 10(1) (2018), pp. 73-83.
[8] H. Ghahramani, Linear maps on group algebras determined by the action of the derivations or anti-derivations on a set of orthogonal elements, Result. Math., 73.4 (2018), pp. 1-14.
[9] H. Ghahramani and Z. Pan, Linear maps on $*$-algebras acting on orthogonal elements like derivations or anti-derivations, Filomat, 32.13 (2018), pp. 4543-4554.
[10] J.R. Giles, Classes of semi-inner-product spaces, Trans. Am. Math. Soc., 129.3 (1967), pp. 436-446.
[11] J. He, J. Li, and D. Zhao. Derivations, local and $2-$local derivations on some algebras of operators on Hilbert C$^*-$modules, Mediterr. J. Math., 14.6 (2017), pp. 1-11.
[12] T. Husain and B.D. Malviya, On semi-inner product spaces, II, Colloq. Math. Institute of Mathematics Polish Academy of Sciences, 1973.
[13] I. Kaplansky, Modules over operator algebras, Am. J. Math., 75.4 (1953), pp. 839-858.
[14] S. Kim and J. Kim, Local automorphisms and derivations on $M_n$, Proc. Am. Math. Soc., 132.5 (2004), pp. 1389-1392.
[15] G. Lumer, Semi-inner-product spaces, Trans. Am. Math. Soc., 100.1 (1961), pp. 29-43.
[16] L. Molnar, Local automorphisms of operator algebras on Banach spaces, Proc. Am. Math. Soc., 131.6 (2003), pp. 1867-1874.
[17] G.J. Murphy, C$^*$-Algebras and Operator Theory, Elsevier Science, 1990.
[18] B. Nath, On a generalization of semi-inner product spaces, Math. J. Okayama Univ., 15.1 (1971), pp. 1-6. 
[19] W.L. Paschke, Inner product modules over B$^*$-algebras, Trans. Am. Math. Soc., 182 (1973), pp. 443-468.
[20] N.C. Phillips and N. Weaver, Modules with norms which take values in a C$^*$-algebra, Pac. J. Appl. Math., 185.1 (1998), pp. 163-181.
[21] M.A. Rieffel, Induced representations of C$^*$-algebras, Adv. Math., 13.2 (1974), pp. 176-257.
[22] P. Semrl, Local automorphisms and derivations on $B(H)$, Proc. Am. Math. Soc., 125.9 (1997), pp. 2677-2680.
[23] A.M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Am. Math. Soc., 24.1 (1970), pp. 209-214.