Derivations of non-separable C1-algebras |
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Authors: | Charles A Akemann BE Johnson |
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Institution: | University of California, Department of Mathematics, Santa Barbara, California 93106 U.S.A.;School of Mathematics, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, England |
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Abstract: | The question of which C1-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C1-tensor product of a von Neumann algebra and an abelian C1-algebra has only inner derivations. Other special types of C1-algebras are shown to have only inner derivations as well such as the C1-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C1-algebra having only inner derivations. Derivations from a smaller C1-algebra into a larger one are also considered, and this concept is generalized to include derivations between C1-algebras connected by a 1-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C1-algebra which converges to zero (in norm) when restricted to any abelian C1-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general. |
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