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SVEP for multipliers on a faithful commutative Banach algebra

Author(s): Janko Bracic; Martin Jesenko
Journal: Proc. Amer. Math. Soc. 135 (2007), 3181-3185.
MSC (2000): Primary 47B40, 47B48
Posted: May 10, 2007
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Abstract: We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.

References:

1.: C. Benhida, E. H. Zerouali and H. Zguitti, Spectra of upper triangular operator matrices, Proc. Amer. Math. Soc. 133(2005), 3013-3020. MR 2159780 (2006i:47005)
2.: R. Larsen, An Introduction to the Theory of Multipliers, Springer, 1971. MR 0435738 (55:8695)
3.: K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, Clarendon, Oxford, 2000. MR 1747914 (2001k:47002)
4.: V. Müller, Adjoining inverses to noncommutative Banach algebras and extensions of operators, Studia Math. 91(1988), no. 1, 73-77. MR 0957286 (90a:46117)


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