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Continuity of Actions of Groups and Semigroups on Banach Spaces
Author(s) -
Brown Lawrence G.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001058
Subject(s) - mathematics , banach space , pure mathematics , subspace topology , invariant (physics) , action (physics) , locally compact space , group (periodic table) , discrete mathematics , mathematical analysis , physics , quantum mechanics , mathematical physics
It is shown that if a locally compact group acts isometrically on a Banach space X leaving a closed subspace M invariant, and if the induced actions on M and X / M are strongly continuous, then the action on X is strongly continuous. Since this may be of interest for one‐parameter semigroups, similar results are proved for actions of suitable topological semigroups. Other generalizations are given for (suitable) non‐isometric actions, non‐locally compact groups, and non‐Banach spaces; corollaries concerning 1‐cocycles and uniformly continuous actions are given.