Idempotents of norm one and Banach algebra representations of compact groups
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- by N. J. Kalton and G. V. Wood PDF
- Proc. Amer. Math. Soc. 55 (1976), 361-366 Request permission
Abstract:
Let $G$ be a finite group of order $n$ and let $A$ be a (real or complex) Banach algebra. Rudin and Schneider [3] ask whether a mapping $f:G \to A$ satisfying $||f(x)|| = 1$ and $f(x) = (1/n){\Sigma _{y \in G}}f(x{y^{ - 1}})f(y)$ is necessarily a homomorphism (Question 1, p. 602). They give an affirmative answer if $A$ is either commutative and semisimple or strictly convex. Here, we prove this result for general Banach algebras, and at the same time prove the natural generalization to compact groups. This allows us to characterize norm one idempotents in generalized group algebras.References
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- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- Walter Rudin and Hans Schneider, Idempotents in group rings, Duke Math. J. 31 (1964), 585–602. MR 167853
- Geoffrey V. Wood, Homomorphisms of group algebras, Duke Math. J. 41 (1974), 255–261. MR 350323
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 55 (1976), 361-366
- DOI: https://doi.org/10.1090/S0002-9939-1976-0397311-1
- MathSciNet review: 0397311