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One interesting example is A discontinuous homomorphism from C(X) without CH, by W. Hugh Woodin, which begins with the following introduction.

Suppose that X is an infinite compact Hausdorff space and let C(X) be the algebra of continuous real-valued functions on X. Then C(X) is a commutative Banach algebra relative to the supnorm: || f || = sup{ f(p) | p ∈ X }. A well-known question of I. Kaplansky posed around 1947 asks whether every algebra homomorphism of C{X) into a Banach algebra B is necessarily continuous.

There is a discontinuous homomorphism of C(X) if and only if there is a discontinuous homomorphism of C(X, C), the C*-algebra of continuous complexvalued functions on X. We prefer to deal with the real case; some of the references adopt the complex view. The question is now known to be independent of the axioms of set theory, ZFC. H. G. Dales [1] and J. Esterle [3] independently constructed discontinuous homomorphisms of C{X) for any infinite space X assuming the Continuum Hypothesis, CH. About the same time R. Solovay [7] proved that it is relatively consistent with ZFC that every homomorphism of C(X) for any space X is necessarily continuous. Solovay's result was improved [8] fairly soon thereafter, to obtain the relative consistency with ZFC -I- + Martin's Axiom (ZFC + MA) that every homomorphism of C(X) for any space X is continuous. We refer the reader to [2] for an exposition of the latter result concerning MA, historical points and related results. After these results several questions remained. This paper is concerned with the question of whether the existence of a discontinuous homomorphism of C{X) is possible given the failure of the Continuum Hypothesis.

A standard method to obtain the consistency of a proposition with the negation of the Continuum Hypothesis (-> CH) ¬CH) when a proof of the proposition assuming CH is known is to attempt to use MA + -> CH ¬CH in place of CH and prove the proposition. However by the result indicated above, this approach will not succeed. The main theorem of this paper, formulated using the terminology of forcing, is the following.

THEOREM. Assume CH. Let P be the Cohen partial order for adding ω2 Cohen reals. Then in VP there exists a discontinuous homomorphism of C(X)for every infinite compact Hausdorff space X.

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One interesting example is A discontinuous homomorphism from C(X) without CH, by W. Hugh Woodin, which begins with the following introduction.

Suppose that X is an infinite compact Hausdorff space and let C(X) be the algebra of continuous real-valued functions on X. Then C(X) is a commutative Banach algebra relative to the supnorm: || f || = sup{ f(p) | p ∈ X }. A well-known question of I. Kaplansky posed around 1947 asks whether every algebra homomorphism of C{X) into a Banach algebra B is necessarily continuous.

There is a discontinuous homomorphism of C(X) if and only if there is a discontinuous homomorphism of C(X, C), the C*-algebra of continuous complexvalued functions on X. We prefer to deal with the real case; some of the references adopt the complex view. The question is now known to be independent of the axioms of set theory, ZFC. H. G. Dales [1] and J. Esterle [3] independently constructed discontinuous homomorphisms of C{X) for any infinite space X assuming the Continuum Hypothesis, CH. About the same time R. Solovay [7] proved that it is relatively consistent with ZFC that every homomorphism of C(X) for any space X is necessarily continuous. Solovay's result was improved [8] fairly soon thereafter, to obtain the relative consistency with ZFC -I- Martin's Axiom (ZFC + MA) that every homomorphism of C(X) for any space X is continuous. We refer the reader to [2] for an exposition of the latter result concerning MA, historical points and related results. After these results several questions remained. This paper is concerned with the question of whether the existence of a discontinuous homomorphism of C{X) is possible given the failure of the Continuum Hypothesis.

A standard method to obtain the consistency of a proposition with the negation of the Continuum Hypothesis (-> CH) when a proof of the proposition assuming CH is known is to attempt to use MA + -> CH in place of CH and prove the proposition. However by the result indicated above, this approach will not succeed. The main theorem of this paper, formulated using the terminology of forcing, is the following.

THEOREM. Assume CH. Let P be the Cohen partial order for adding ω2 Cohen reals. Then in VP there exists a discontinuous homomorphism of C(X)for every infinite compact Hausdorff space X.