Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An order characterization of commutativity for $C^{\ast}$-algebras

Author: Wei Wu
Journal: Proc. Amer. Math. Soc. 129 (2001), 983-987
MSC (2000): Primary 46L05
Published electronically: October 10, 2000
MathSciNet review: 1814137
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate the problem of when a $C^{\ast }$-algebra is commutative through operator-monotonic increasing functions. The principal result is that the function $e^{t}, t\in [0, \infty ),$ is operator-monotonic increasing on a $C^{\ast }$-algebra $\mathcal{A}$ if and only if $\mathcal{A}$ is commutative. Therefore, $C^{\ast }$-algebra $\mathcal{A}$ is commutative if and only if $e^{x+y}=e^{x}e^{y}$ in $\mathcal{A} \dot + \mathbf{C}$ for all positive elements $x, y$ in $\mathcal{A}$.

References [Enhancements On Off] (What's this?)

  • 1. M. J. Crabb, J. Duncan and C. M. McGregor, Characterizations of commutativity for $C^{\ast }$- algebras, Glasgow Math. J. 15 (1974), 172-175. MR 50:14252
  • 2. J. Dixmier, $C^{\ast }$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam, 1977. MR 56:16388
  • 3. R. S. Doran and V. A. Belfi, Characterizations of $C^{\ast }$-algebras, The Gelfand-Naimark Theorems, Marcel Dekker, Inc., New York, 1986. MR 87k:46115
  • 4. J. Duncan and P. J. Taylor, Norm Inequalities for $C^{\ast }$-algebras, Proc. Roy. Soc. Edinburgh, Sect. A, 75 (1975/76), 119-129. MR 56:12896
  • 5. M. Fukamiya, M. Misonou and Z. Takeda, On order and commutativity of $B^{\ast }$-algebras, T${\hat o}$hoku Math. J. (2)6 (1954), 89-93. MR 16:376
  • 6. F. Hansen and G. K. Pedersen, Jensen's inequality for operators and Lowner's theorem, Math. Ann. 258 (1982), 229-241. MR 83g:47020
  • 7. R. V. Kadison and J. R. Ringrose, Foundamentals of the theory operator algebras, Vol. 1, Elementary theory, Pure and Applied Mathematics, 100, Academic Press, Inc., New York-London, 1983. MR 85j:46099
  • 8. Y. Kato, A characterization of commutative $C^{\ast }$-algebras by normal approximate spectra, Math. Japon. 24 (1979/80), 209-210. MR 81f:46071
  • 9. B. R. Li, Introduction to operator algebras, World Scientific, Singapore, 1992. MR 94b:46083
  • 10. R. Nakamoto, A spectral characterization of commutative $C^{\ast }$-algebras, Math. Japon. 24 (1979/80), 399-400. MR 81h:46074
  • 11. T. Ogasawara, A theorem on operator algebras, J. Sci. Hiroshima Univ. Ser. A 18 (1955), 307-309. MR 17:514
  • 12. S. Sherman, Order in operator algebras, Amer. J. Math. 73 (1951), 227-232. MR 13:47
  • 13. C. F. Skau, Orthogonal measures on the state space of a $C^{\ast }$-algebra, in: Algebras in Analysis (Conference Proceedings, Academic Press, London, 1975), 272-303. MR 54:8302
  • 14. S. Wright, On factor states, Rocky Mountain J. Math. 12 (1982), 569-579. MR 83j:46075

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L05

Retrieve articles in all journals with MSC (2000): 46L05

Additional Information

Wei Wu
Affiliation: Institute of Mathematics, Academia Sinica, Beijing 100080, China
Address at time of publication: Department of Mathematics, East China Normal University, Shanghai 200062, China

Keywords: Commutativity for $C^{\ast }$-algebras, operator-monotonic increasing function, positive element
Received by editor(s): November 4, 1998
Received by editor(s) in revised form: June 4, 1999
Published electronically: October 10, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society