By Earl R. Berkson, N. Tenney Peck, J. Jerry Uhl

In the course of the educational 12 months 1986-7, the collage of Illinois used to be host to a symposium on mathematical research which was once attended through the various top figures within the box. This e-book arises out of this specific yr and lays emphasis at the synthesis of recent and classical research. The contributed articles via the individuals hide the gamut of mainstream subject matters. This e-book should be necessary to researchers in mathematical research.

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240-241]). class UMD contains the space for 1 < P < p (c) The LP-spaces associated with an arbitrary measure as well as their non-commutative counterparts, ~, including the von Neumann-Schatten p-classes for P in the same range. Also, the class UMD is closed under the formation of dual spaces, quotient spaces, and subspaces, and each UMD space is reflexive (in fact, super-reflexive). For more detailed background information and further references, see [4,5,8]. v We can now obtain vector-valued versions of Steckin's theorems.

Then § G(L) that = 0 } , define = al 9 , into ACQ(TP) f) ( t € IR ) . =T §: an a l g e b r a h o m o m o r p h i s m + and such is an isometric a l g e b r a isomorphism of Let h o m o m o r p h i s m of f(a differentiable setting f —» ACQ{IR) (0,2Tr) { G € ACCIT) : flf(t) = The m a p p i n g »(X) . i n f i n i t e ly o n to the o p e n interval Given I so that with || —» SD(X) by » ( X ) AC(T) is an + ( is i d e n t i t y - p r e s e r v i n g w i t h a € (C ^q II , ACQCIT) a l g e b ra - ^ ' Extend d e f i n i ng f li $ li < m a x € ACQCIT) {K,l} .

20)]). 4) below) of Steckin's multiplier theorems [12]. tools, the above results on UMD spaces follow easily. 4] can be obtained from the vector-valued version, so that the latter can be viewed as a generalization of the former. As usual, IR Z, ~, N, and T denote, respectively, the reals, the complexes, the integers, the positive integers, and the circle group. Total variation is abbreviated "var". a compact interval in IR, BV(J) J, AC(J) =I J. algebra of all functions fIt) = f(e it ) "T f: J =" f (b ) I is of bounded variation ~ ~ + var ( f , J) Similarly, f: T ~ ~ BV(T) , and f AC(T) such that algebra of all functions f: IR denotes the Banach for which the function is of bounded variation on "[O,2n] of those functions f f [a,b] is the closed subalgebra consisting of the absolutely continuous functions on " = with norm II f II J and J denotes the Banach algebra, under pointwise operations, of all functions on If [O,2n] , with norm is the closed subalgebra consisting f ~ ~ is in AC([0,2n]) .