Abstract
We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure μ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Lie algebra” of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L 2(ν)-closure of holomorphic polynomials by their values on the Cameron–Martin subgroup.
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Acknowledgments
We are grateful to Professor Malliavin whose question during a workshop at the Hausdorff Institute (Bonn, Germany) led us to include a section on a holomorphic chaos expansion.
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B. K. Driver’s research was supported in part by NSF Grants DMS-0504608 and DMS-0804472 and the Miller Institute at the University of California, at Berkeley.
M. Gordina’s research was supported in part by NSF Grant DMS-0706784 and the Humboldt Foundation Research Fellowship.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Driver, B.K., Gordina, M. Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups. Probab. Theory Relat. Fields 147, 481–528 (2010). https://doi.org/10.1007/s00440-009-0213-y
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DOI: https://doi.org/10.1007/s00440-009-0213-y