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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Projective limits in harmonic analysis

Author: William A. Greene
Journal: Trans. Amer. Math. Soc. 209 (1975), 119-142
MSC: Primary 22D15; Secondary 43A95
MathSciNet review: 0376952
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Abstract: A treatment of induced transformations of measures and measurable functions is presented. Given a diagram $\varphi :G \to H$ in the category of locally compact groups and continuous proper surjective group homomorphisms, functors are produced which on objects are given by $G \to {L^2}(G),{L^1}(G)$, $M(G),W(G)$, denoting, resp., the ${L^2}$-space, ${L^1}$-algebra, measure algebra, and von Neu mann algebra generated by left regular representation of ${L^1}$ on ${L^2}$. All functors but but the second are shown to preserve projective limits; by example, the second is shown not to do so. The category of Hilbert spaces and linear transformations of norm $\leqslant 1$ is shown to have projective limits; some propositions on such limits are given. Also given is a type and factor characterization of projective limits in the category of ${W^ \ast }$-algebras and surjective normal $\ast$-algebra homomorphisms.

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Keywords: Functor, category, limit, projective limit, categorical limit preservation, locally compact group, Haar measure, convolution measure algebra, <IMG WIDTH="29" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img15.gif" ALT="${L^p}$">-space, Banach space, Hilbert space, <!– MATH ${C^ \ast }$ –> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="${C^ \ast }$">-algebra, <!– MATH ${W^ \ast }$ –> <IMG WIDTH="37" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${W^ \ast }$">-algebra
Article copyright: © Copyright 1975 American Mathematical Society