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Discontinuous Homomorphisms from Non‐Commutative Banach Algebras
Author(s) -
Dales H. G.,
Runde V.
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397002981
Subject(s) - mathematics , homomorphism , commutative property , banach algebra , pure mathematics , algebra homomorphism , eberlein–šmulian theorem , banach space , algebra over a field , discrete mathematics , lp space
In the 1970s, a question of Kaplansky about discontinuous homomorphisms from certain commutative Banach algebras was resolved. Let A be the commutative C *‐algebra C (Ω), where Ω is an infinite compact space. Then, if the continuum hypothesis (CH) be assumed, there is a discontinuous homomorphism from C (Ω) into a Banach algebra [ 2 , 7 ]. In fact, let A be a commutative Banach algebra. Then (with (CH)) there is a discontinuous homomorphism from A into a Banach algebra whenever the character space Φ A of A is infinite [ 3 , Theorem 3] and also whenever there is a non‐maximal, prime ideal P in A such that | A / P | = 2 ℵ 0[ 4 , 8 ] . (It is an open question whether or not every infinite‐dimensional, commutative Banach algebra A satisfies this latter condition.) 1991 Mathematics Subject Classification 46H40.