C*-Algebras with Dense Nilpotent Subalgebras

In a beautiful result, Herrero (D. A. Herrero, ‘Normallimits of nilpotent operators’, Indiana Univ. Math. J.23 (1973/74) 1097–1108) showed that a normal operatoron l² lies in the closure of the set of nilpotent operatorsif and only if its spectrum is connected and contains zero.In the quest for an automatic continuity result for algebrahomomorphisms between C* -algebras, Dales showed that, if adiscontinuous algebra homomorphism : A u exists between C*-algebrasA and u, and if (A) is dense in u, then there is a C*-algebrau₂ with a dense subalgebra N u² such that every x N is quasinilpotent(see p. 685 of H. G. Dales, Banach algebras and automatic continuity,London Mathematical Society Monographs 24, Oxford UniversityPress, 2001). (A discontinuous homomorphism ₂: A₂ u₂ can bedefined with the same basic properties as , but the revisedtarget space u₂ has a dense subalgebra consisting of quasinilpotentelements.) As remarked by Dales, no such C*-algebra was thenknown; but here we present one. Indeed, using the full powerof Herrero's result, one may arrange that every x N is nilpotent.The C*-algebra is constructed in a ‘neat’ way; itis most naturally constructed as a non-separable, concrete C*-algebraof operators on a separable Hilbert space K but one can arrangethat the algebra u itself be separable if desired. 2000 MathematicsSubject Classification 47C15, 46H40 (primary), 47A10, 46L06,46L05, 46H35 (secondary).