Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups |
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Authors: | Bruce K Driver Maria Gordina |
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Institution: | 1. Department of Mathematics, 0112, University of California, San Diego, La Jolla, CA, 92093-0112, USA 2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA
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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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