Compact endomorphisms and closed ideals in Banach algebras

Grabiner, Sandy

doi:10.1090/S0002-9939-1984-0760943-7

Compact endomorphisms and closed ideals in Banach algebras
HTML articles powered by AMS MathViewer

by Sandy Grabiner PDF

Proc. Amer. Math. Soc. 92 (1984), 547-548 Request permission

Abstract:

Every infinite-dimensional Banach algebra with a nonzero compact endomorphism has a proper closed nonzero two-sided ideal. When the algebra is commutative, the ideal is also an ideal in the multiplier algebra.

References

J. Esterle, Quasimultipliers, representations of $H^{\infty }$, and the closed ideal problem for commutative Banach algebras, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981) Lecture Notes in Math., vol. 975, Springer, Berlin, 1983, pp. 66–162. MR 697579, DOI 10.1007/BFb0064548
C. K. Fong, E. A. Nordgren, M. Radjabalipour, H. Radjavi, and P. Rosenthal, Extensions of Lomonosov’s invariant subspace theorem, Acta Sci. Math. (Szeged) 41 (1979), no. 1-2, 55–62. MR 534499
Sandy Grabiner, Derivations and automorphisms of Banach algebras of power series, Memoirs of the American Mathematical Society, No. 146, American Mathematical Society, Providence, R.I., 1974. MR 0415321
Sandy Grabiner, Operator ranges and invariant subspaces, Indiana Univ. Math. J. 28 (1979), no. 5, 845–857. MR 542341, DOI 10.1512/iumj.1979.28.28059
Herbert Kamowitz, Compact endomorphisms of Banach algebras, Pacific J. Math. 89 (1980), no. 2, 313–325. MR 599123
V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Funkcional. Anal. i Priložen. 7 (1973), no. 3, 55–56 (Russian). MR 0420305
Carl Pearcy and Allen L. Shields, A survey of the Lomonosov technique in the theory of invariant subspaces, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 219–229. MR 0355639