TY - GEN

T1 - Theory of valuations on manifolds, IV. New properties of the multiplicative structure

AU - Alesker, S.

PY - 2007

Y1 - 2007

N2 - This is the fourth part in the series of articles [A4], [A5], [AF] (see also [A3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations is an automorphism of the algebra of valuations. Then an integration functional on valuations with compact support is introduced, and a property of selfduality of valuations is proved. Next a space of generalized valuations is defined, and some basic properties of it are proved. Finally a canonical imbedding of the space of constructible functions on a real analytic manifold into the space of generalized valuations is constructed, and various structures on valuations are compared with known structures on constructible functions.

AB - This is the fourth part in the series of articles [A4], [A5], [AF] (see also [A3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations is an automorphism of the algebra of valuations. Then an integration functional on valuations with compact support is introduced, and a property of selfduality of valuations is proved. Next a space of generalized valuations is defined, and some basic properties of it are proved. Finally a canonical imbedding of the space of constructible functions on a real analytic manifold into the space of generalized valuations is constructed, and various structures on valuations are compared with known structures on constructible functions.

UR - http://www.scopus.com/inward/record.url?scp=34247590980&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-72053-9_1

DO - 10.1007/978-3-540-72053-9_1

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AN - SCOPUS:34247590980

SN - 3540720529

SN - 9783540720522

T3 - Lecture Notes in Mathematics

SP - 1

EP - 44

BT - Geometric Aspects of Functional Analysis

PB - Springer Verlag

ER -