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A note on the Gelfand-Mazur Theorem

Author(s): S. J. Bhatt; D. J. Karia; S. H. Kulkarni; M. E. Shimpi
Journal: Proc. Amer. Math. Soc. 126 (1998), 2999-3005.
MSC (1991): Primary 46K15, 46H20
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Abstract: Three Gelfand-Mazur type theorems are proved. One of these provides a $C^*$-property analogue of Zalar's recent generalizations of the Froelich-Ingelstam-Smiley Theorems concerning unital multiplication in Hilbert spaces; the second illustrates that the assumption in Kaplansky's version of the Gelfand-Mazur Theorem can be weakened in the presence of a $C^*$-norm; whereas the third provides a real analogue of a result due to Srinivasan.

References:

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S. J. Bhatt, Some remarks on hermitian Banach $^*$-algebras, Indian J. Pure Appl. Math. 26 (2) (1995), 131-142. MR 96e:46065

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F. F. Bonsall, and J. Duncan, Complete Normed Algebras, Springer-Verlag, Berlin, Heidelberg, New York, 1973. MR 54:11013

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J. Froelich, Unital multiplication on a Hilbert space, Proc. Amer. Math. Soc. 117 (1993), 757-759. MR 93d:46081

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M. C. Garcia and A. R. Palacios, A new simple proof of the Gelfand-Mazur Theorem, Proc. Amer. Math. Soc. 123 (9) (1995), 2663-2666. MR 95k:46077

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J. Heinze, A remark on a paper of V. K. Srinivasan, Bull. Austral. Math. Soc. 22 (1980), 153-154. MR 82a:46045

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V. K. Srinivasan, On Some Gelfand-Mazur like theorems in Banach algebras, Bull. Austral. Math. Soc. 20 (1979), 211-215; Corrigenda, ibid 23 (1981), 497-480. MR 81e:46029a; MR 82h:46067

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Additional Information:

S. J. Bhatt
Affiliation: Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India

S. H. Kulkarni
Affiliation: Department of Mathematics, Indian Institute of Technology, Madras, 600 036, India

M. E. Shimpi
Affiliation: Department of Mathematics, BVM Engineering College, Vallabh Vidyanagar-388 120, Gujarat, India

DOI: 10.1090/S0002-9939-98-04658-9
PII: S 0002-9939(98)04658-9
Keywords: Real *-algebra, Pythagorean norm, Gelfand-Mazur Theorem, Froelich-Ingelstam-Smiley Theorem
Received by editor(s): September 23, 1996
Received by editor(s) in revised form: March 12, 1997
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1998, American Mathematical Society

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