A note on the Gelfand-Mazur Theorem

Authors:
S. J. Bhatt, D. J. Karia, S. H. Kulkarni and M. E. Shimpi

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2999-3005

MSC (1991):
Primary 46K15, 46H20

DOI:
https://doi.org/10.1090/S0002-9939-98-04658-9

MathSciNet review:
1476120

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Abstract | References | Similar Articles | Additional Information

Abstract: Three Gelfand-Mazur type theorems are proved. One of these provides a -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 -norm; whereas the third provides a real analogue of a result due to Srinivasan.

**[B]**S. J. Bhatt,*Some remarks on hermitian Banach -algebras*, Indian J. Pure Appl. Math.**26**(2) (1995), 131-142. MR**96e:46065****[BD]**F. F. Bonsall, and J. Duncan,*Complete Normed Algebras*, Springer-Verlag, Berlin, Heidelberg, New York, 1973. MR**54:11013****[Fr]**J. Froelich,*Unital multiplication on a Hilbert space*, Proc. Amer. Math. Soc.**117**(1993), 757-759. MR**93d:46081****[G]**K. R. Goodearl,*Notes on Real and Complex -Algebras*, Shiv Publ. Co., England, 1987. MR**85d:46079****[GP]**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****[He]**J. Heinze,*A remark on a paper of V. K. Srinivasan*, Bull. Austral. Math. Soc.**22**(1980), 153-154. MR**82a:46045****[S]**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****[Z]**B. Zalar,*On Hilbert space with unital multiplication*, Proc. Amer. Math. Soc.**123**(1995), 1497-1501. MR**95f:46093**

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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:
https://doi.org/10.1090/S0002-9939-98-04658-9

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

Article copyright:
© Copyright 1998
American Mathematical Society