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The Mathematician as a Formalist
 in Truth in Mathematics (H.G. Dales and
, 1998
"... Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millenni ..."
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Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millennium; it would be implausible and perhaps presumptuous to suppose that even the union of the talented and distinguished speakers that have been assembled here in Mussomeli will approach any solution to the problem, or even arrive at a consensus of what a solution would amount to. In the end, it falls to the philosophers, with their professional expertise and training, to carry forward the debate and to move us to a fuller understanding of this subtle and elusive matter. Indeed, we are hearing at this meeting a variety of contributions to the debate from different philosophical points of view; also, there is a good number of recent published contributions to the debate (see (Maddy 1990)
A history of conferences on Banach algebras
"... . We record a history of the thirteen conferences on Banach algebras that have taken place; the first was in Los Angeles in 1974 and the thirteenth is the conference in Blaubeuren recorded in the present volume. 1991 Mathematics Subject Classification: 46H99, 01A65. Preliminaries This article is wr ..."
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. We record a history of the thirteen conferences on Banach algebras that have taken place; the first was in Los Angeles in 1974 and the thirteenth is the conference in Blaubeuren recorded in the present volume. 1991 Mathematics Subject Classification: 46H99, 01A65. Preliminaries This article is written to record  at least from the memories of some of the participants  the history of the sequence of conferences on Banach algebras; the Blaubeuren conference is the 13 th conference in this sequence. We seek to record some mathematical and some organizational facts, and to ruminate on the changing mathematical and organizational scene. In writing this account I have come to realize that my memory is a little wobblier than I had supposed; even if one was present at an event, one cannot necessarily recall accurately all the details. One is led to wonder how reliable is the report of historians, who were not present at the events that they describe. To some extent I have been able t...
Questions on Automatic Continuity
"... . We present a variety of open questions in automatic continuity theory, concentrating on homomorphisms between Banach algebras and derivations from a Banach algebra A into a Banach Abimodule. 1991 Mathematics Subject Classification: 46H40. 1. Introduction In automatic continuity theory, we are c ..."
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. We present a variety of open questions in automatic continuity theory, concentrating on homomorphisms between Banach algebras and derivations from a Banach algebra A into a Banach Abimodule. 1991 Mathematics Subject Classification: 46H40. 1. Introduction In automatic continuity theory, we are concerned with conditions that imply that a linear map between Banach spaces (or more general topological linear spaces) is necessarily continuous. A rather general context in which questions are posed is the following. Let (A; k \Delta k) be a Banach algebra, and let (E; k \Delta k) be a Banach space which is an Abimodule for the maps (a; x) 7! a \Delta x and (a; x) 7! x \Delta a. Then E is a weak Banach Abimodule if the maps x 7! a \Delta x and a 7! x \Delta a on E are continuous for each a 2 A. For example, suppose that A and B are Banach algebras and that ` : A ! B is a homomorphism. Then B is a weak Banach Abimodule for the module operations given by a \Delta b = `(a)b and b \Delta a...
Automatic continuity for Banach algebras
"... ollows easily from the continuity of characters that every homomorphism ` : A ! B from a Banach algebra A into a commutative, semisimple Banach algebra B is continuous. A closely related result is Johnson's uniquenessofnormtheorem: every semisimple Banach algebra has a unique complete algebra n ..."
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ollows easily from the continuity of characters that every homomorphism ` : A ! B from a Banach algebra A into a commutative, semisimple Banach algebra B is continuous. A closely related result is Johnson's uniquenessofnormtheorem: every semisimple Banach algebra has a unique complete algebra norm. For lovely alternative proofs of this theorem, see [1] and [13]. There are nonsemsimple, commutative Banach algebras which have a unique complete algebra norm. For example, this is true of the convolution algebras L (R ; !), where ! is a weight function on R (see [4], x5.2). On the other hand, there are even commutative Banach algebras with a onedimensional (Jacobson) radical which do not have a unique complete algebra norm (see [4], x5.1). Nevertheless there are striking open questions in this area: we do not know whether a commutative Banach algebra which is an integral domain necessarily has a unique complete algebra norm; the question is also open for Banach algebras with
URL: www.emis.de/journals/AFA/ KAPLANSKY’S AND MICHAEL’S PROBLEMS: A SURVEY
"... Abstract. I. Kaplansky showed in 1947 that every submultiplicative norm ‖. ‖ on the algebra C(K) of complex–valued functions on an infinite compact space K satisfies ‖f ‖ ≥ ‖f‖K for every f ∈ C(K), where ‖f‖K = maxt∈Kf(t) denotes the standard norm on C(K). He asked whether all submultiplicative n ..."
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Abstract. I. Kaplansky showed in 1947 that every submultiplicative norm ‖. ‖ on the algebra C(K) of complex–valued functions on an infinite compact space K satisfies ‖f ‖ ≥ ‖f‖K for every f ∈ C(K), where ‖f‖K = maxt∈Kf(t) denotes the standard norm on C(K). He asked whether all submultiplicative norms ‖. ‖ were in fact equivalent to the standard norm (which is obviously true for finite compact spaces), or equivalently, whether all homomorphisms from C(K) into a Banach algebra were continuous. This problem turned out to be undecidable in ZFC, and we will discuss here some recent progress due to Pham and open questions concerning the structure of the set of nonmaximal prime ideals of C(K) which are closed with respect to a discontinuous submultiplicative norm on C(K) when the continuum hypothesis is assumed. We will also discuss the existence of discontinuous characters on Fréchet algebras (Michael’s problem), a long standing problem which remains unsolved. The Mittag–Leffler theorem on inverse limits of complete metric spaces plays an essential role in the literature concerning both problems. 1. Introduction and