Get An Introduction to the Theory of Multipliers PDF

By Ronald Larsen

ISBN-10: 3642650309

ISBN-13: 9783642650307

ISBN-10: 3642650325

ISBN-13: 9783642650321

When I first thought of writing a publication approximately multipliers, it was once my purpose to provide a average sized monograph which lined the idea as an entire and which might be available and readable to a person with a uncomplicated wisdom of practical and harmonic research. I quickly learned, besides the fact that, that this type of target couldn't be attained. This recognition is obvious within the preface to the initial model of the current paintings which was once released within the Springer Lecture Notes in arithmetic, quantity one hundred and five, and is much more acute now, after the revision, enlargement and emendation of that manuscript had to produce the current quantity. hence, as sooner than, the remedy given within the following pages is eclectric instead of definitive. the alternative and presentation of the themes is unquestionably no longer targeted, and displays either my own personal tastes and inadequacies, in addition to the need of limiting the publication to an inexpensive measurement. all through i've got given certain emphasis to the func­ tional analytic points of the characterization challenge for multipliers, and feature, commonly, in basic terms awarded the commutative model of the speculation. i've got additionally, optimistically, supplied too many info for the reader instead of too few.

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Extra info for An Introduction to the Theory of Multipliers

Example text

Therefore Ox is compact for all xEA and I(A) is compact. 0 As will be seen shortly, the next criterion for the compactness of I(A) will be the most useful one for our investigation. However, before giving it, we wish to make another definition. An element XEA is said to be almost invariant if Ox spans a finite dimensional subspace of A. The set of all almost invariant elements in A will be denoted by D(A). Clearly D(A) is an ideal in A since, in general, Oyx= yOx. 2. Let A be a semi-simple commutative Banach algebra.

1. Let A be a semi-simple commutative Banach algebra. It(A). It(A) is not an isometry. However if A is a supremum norm algebra, that is, Ilxll = Ilxll oo for all xEA, then the isomorphism is isometric. 3. Let A be a semi-simple commutative supremum norm algebra. It(A) is such that (Txf = qJ for all XE A then I Til = IlqJll 00' x Proof. 2 we know that I qJ I 00 ~ I Til. On the other hand I Til = sup I Txll = sup IlqJxll ~ IlqJll Ilxll=l Ilxll oo =l 00 00' Hence IITII=llqJlloo· 0 Thus we see that for semi-simple commutative Banach algebras we can consider M(A) as a space of bounded continuous functions defined on the regular maximal ideal space of A.

Thus 111111 = IISII· Combining all of the previous arguments we see that F defines an isometric isomorphism of M[I(A)J/N onto M(A). Next we wish to show that A is isometrically isomorphic to the norm closure of y(L1 [I(A)]}. We begin by showing that the norm closure in M(A) of F(L 1[I(A)]) is equal to {LxlxEA}. Suppose fED(L1[I(A)]). Then, since bT for each TEI(A) defines an isometric multiplier for L1 [I(A)], we see that the set {(F f) T= F(f *bT)ITEI(A)} spans a finite dimensional space X(f).

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An Introduction to the Theory of Multipliers by Ronald Larsen


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