Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem
Emanating from the theory of C*-algebras and actions of tori theoren, the problems discussed here are outgrowths of random walk problems on lattices. An AGL (d,Z)-invariant (which is a partially ordered commutative algebra) is obtained for lattice polytop
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1282
David E. Handelman
Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem
Jerg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1282
David E. Handelman
Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem
Jerg New York London Paris Tokyo
Author
David E. Handelman Mathematics Department, University of Ottawa Ottawa, Ontario K 1N 6N5, Canada
Mathematics Subject Classification (1980): 06F 25, 13B99, 19A99, 19K 14, 46L99, 52A43, 60G50 ISBN 3-540-18400-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387 -18400- 7 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
POSITIVE POLYNOMIALS, CONVEX INTEGRAL POLYTOPES, AND A RANDOM WALK PROBLEM
This monograph concerns itself with results and interconnections in a number of areas; these include positive polynomials, a class of special random walk problems on the lattice Zd (d an integer), convex integral polytopes (that is, convex polytopes in Rd all of whose vertices are lattice points), reflection groups, and commutative algebra. Techniques include those of functional analysis (especially Choquet theory), ordered rings, commutative algebra, and convex analysis. The central problems arise from special actions of tori on C*-algebras, and many of the results in the other areas yield results back at the C* level; the translation is implemented by means of ordered Ko (of the fixed point C*-algebras). These connections are (roughly) described in Table 1. Since the techniques and results deal with a number of disparate fields, we shall develop the material in each area in considerable detail, as not everyone will be familiar with all of the topics discussed. This prospectus is intended to outline the various interconnections; topics which may not be familiar to the reader will be introduced in the body of the text. In this prospectus I am trying to make the case that there is a lot of interesting mathematics occurring here, and that the scope for further research is vast. The motivating problem arises from the classification and description of invariants arising from "xerox" type actions of tori on C*-algebras. Specifically, let 1t:T
U(n,C) be an n-
dimensional representation of the d-torus T; let A = ® MnC (n fixed) be the infinite tensor product of n x n matrix algebras, and define a:
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