Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 12 (2017), 203 -- 217

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This work is licensed under a Creative Commons Attribution 4.0 International License.

NORMED ALGEBRAS AND THE GEOMETRIC SERIES TEST

Robert Kantrowitz and Michael M. Neumann

Abstract. The purpose of this article is to survey a class of normed algebras that share many central features of Banach algebras, save for completeness. The likeness of these algebras to Banach algebras derives from the fact that the geometric series test is valid, whereas the lack of completeness points to the failure of the absolute convergence test for series in the algebra. Our main result is a compendium of conditions that are all equivalent to the validity of the geometric series test for commutative unital normed algebras. Several examples in the final section showcase some incomplete normed algebras for which the geometric series test is valid, and still others for which it is not.

2010 Mathematics Subject Classification: 46-02; 46H05; 46H20.
Keywords: Normed algebra, Banach algebra, geometric series.

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References

  1. G. R. Allan, Introduction to Banach Spaces and Algebras, Oxford University Press, Oxford, 2011. MR2761146(2012j:46001). Zbl 1220.46001.

  2. T. M. Apostol, Mathematical Analysis, 2nd ed., Addison-Wesley, Reading, MA, 1974. MR0344384(49 #9123). Zbl 0309.26002.

  3. B. Bollobás, Linear Analysis: An Introductory Course, Cambridge University Press, Cambridge, 1990. MR1087297(92a:46001). Zbl 0753.46002.

  4. F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. MR0423029(54 #11013). Zbl 0271.46039.

  5. H. G. Dales, Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000. MR1816726(2002e:46001). Zbl 0981.46043.

  6. R. Fuster and A. Marquina, Geometric series in incomplete normed algebras, Amer. Math. Monthly 91 (1984), 49--51. MR0729192(85g:46059). Zbl 0553.46032.

  7. T. W. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, NJ, 1969. MR0410387(53 #14137). Zbl 0213.40401.

  8. M. Haralampidou, Annihilator topological algebras, Portugal. Math. 51 (1994), 147-162. MR1281963(95f:46076). Zbl 0806.46051.

  9. M. Haralampidou, On the Krull property in topological algebras, Comment. Math. (Prace Mat.) 46 (2006), 141--162. MR2287681(2007i:46045). Zbl 1180.46035.

  10. R. Kantrowitz and M. Neumann, Another face of the Archimedean property, College Math. J. 46 (2015), 139--141. MR3361762.

  11. R. Kantrowitz and M. M. Neumann, More of Dedekind: his series test in normed spaces, Int. J. Math. Math. Sci. 2016, Art. ID 2508172, 3 pp. MR3510935.

  12. R. Kantrowitz and M. M. Neumann, Completeness of ordered fields and a trio of classical series tests, Abstr. Appl. Anal. 2016, Art. ID 6023273, 6 pp. MR3574251.

  13. I. Kaplansky, Topological rings, Amer. J. Math. 69 (1947), 153--183. MR0019596(8,434b). Zbl 0034.16604.

  14. A. Mallios, Topological Algebras. Selected Topics, North-Holland, Amsterdam, 1986. MR0857807(87m:46099). Zbl 0597.46046.

  15. T. W. Palmer, Spectral algebras, Rocky Mountain J. Math. 22 (1992), 293--328. MR1159960(93d:46079). Zbl 0790.46038.

  16. T. W. Palmer, Banach Algebras and The General Theory of *-Algebras, Vol. I: Algebras and Banach Algebras, Cambridge University Press, Cambridge, 1994. MR1270014(95c:46002). Zbl 0809.46052.

  17. J. G. Propp, Real analysis in reverse, Amer. Math. Monthly 120 (2013), 392--408. MR3035440. Zbl 1305.12002.

  18. C. E. Rickart, Book Review: Complete normed algebras, Bull. Amer. Math. Soc. 81 (1975), 514--522.

  19. C. E. Rickart, General Theory of Banach Algebras, Krieger, New York, 1960. MR0115101(22 #5903). Zbl 0095.09702.

  20. K. Saxe, Beginning Functional Analysis, Springer, New York, 2002. MR1871419(2002m:00003). Zbl 1002.46001.

  21. D. Singh, The spectrum in a Banach algebra, Amer. Math. Monthly 113 (2006), 756--758. MR2256536(2007c:46048). Zbl 1146.46022.

  22. Y. Tsertos, A characterization of Q-algebras, Functional analysis, approximation theory and numerical analysis, 277-280, World Sci. Publ., River Edge, NJ, 1994. MR1298668(95i:46078). Zbl 0876.46035.

  23. B. Yood, Homomorphisms on normed algebras, Pacific J. Math. 8 (1958), 373--381. MR0104164(21 #2924). Zbl 0084.33601.



Robert Kantrowitz
Hamilton College
198 College Hill Road Clinton, N.Y. 13323, USA.
e-mail: rkantrow@hamilton.edu

Michael M. Neumann
Mississippi State University, USA
e-mail: neumann@math.msstate.edu

http://www.utgjiu.ro/math/sma