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Projections in Real Banach Algebras

Projections in Real Banach Algebras BERNARD AUPETIT Let A be a real Banach algebra. We denote by P the set of projections of A, that is P = {P £ A : p = p). The set P is closed, but in general it is not connected. In the case of complex Banach algebras, J. Zemanek [4] has shown that P is locally arcwise connected; more precisely, if e and / are in the same connected component of P, then xi Xn x xi there exist x ...,x in A such that / = e ... e ee~ "... e~ and the formula u n tXl tx tx lM 0( 0 = e .. . e "ee~ " ...e~ for 0 ^ t ^ 1 defines an analytic arc in P going from e to / . But the proof of J. Zemanek cannot be adapted to the real case. The aim of this short note is to prove the same result in the real case, by using an argument of Z. V. Kovarik [2] and some ingenious devices. First of all we may suppose that A has a unit because if this is not true we use A , the algebra with unit http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Projections in Real Banach Algebras

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/13.5.412
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Abstract

BERNARD AUPETIT Let A be a real Banach algebra. We denote by P the set of projections of A, that is P = {P £ A : p = p). The set P is closed, but in general it is not connected. In the case of complex Banach algebras, J. Zemanek [4] has shown that P is locally arcwise connected; more precisely, if e and / are in the same connected component of P, then xi Xn x xi there exist x ...,x in A such that / = e ... e ee~ "... e~ and the formula u n tXl tx tx lM 0( 0 = e .. . e "ee~ " ...e~ for 0 ^ t ^ 1 defines an analytic arc in P going from e to / . But the proof of J. Zemanek cannot be adapted to the real case. The aim of this short note is to prove the same result in the real case, by using an argument of Z. V. Kovarik [2] and some ingenious devices. First of all we may suppose that A has a unit because if this is not true we use A , the algebra with unit

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1981

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