TAYLOR MAP ON GROUPS ASSOCIATED WITH A II<sub>1</sub>-FACTOR

Gordina, Maria

doi:10.1142/s0219025702000730

Cited by 9 publications

(21 citation statements)

References 9 publications

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“…In [1,5,12,13,22,23], the analogous Taylor map was also surjective. Section 5 is devoted to proving that our Taylor map is surjective when G is a simply connected graded Lie group.…”

Section: Theorem 7 (Corollary 33) For Any Complex Lie Groupmentioning

confidence: 99%

The Taylor map on complex path groups

Cecil

2008

Journal of Functional Analysis

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Let W(G) denote the path group of an arbitrary complex connected Lie group. The existence of a heat kernel measure ν t on W(G) has been shown in [M. Cecil, B.K. Driver, Heat kernel measure on loop and path groups, preprint, http://www.math.uconn.edu/~cecil/papers/p2.pdf; Infin. Dimens. Anal. Quantum Probab. Relat. Top., submitted for publication]. The present work establishes an isometric map, the Taylor map, from the space of L 2 (ν t )-holomorphic functions on W(G) to a subspace of the dual of the universal enveloping algebra of Lie(H (G)), where H (G) is the Lie subgroup of finite energy paths. This map is shown to be surjective in the case where G is a simply connected graded Lie group.

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“…In [1,5,12,13,22,23], the analogous Taylor map was also surjective. Section 5 is devoted to proving that our Taylor map is surjective when G is a simply connected graded Lie group.…”

Section: Theorem 7 (Corollary 33) For Any Complex Lie Groupmentioning

confidence: 99%

The Taylor map on complex path groups

Cecil

2008

Journal of Functional Analysis

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“…Gordina [10][11][12] considered the Taylor isomorphism in the context of Hilbert-Schmidt groups, while Cecil [2] considered the Taylor isomorphism for path groups over stratified Lie groups. The nilpotentcy of the Heisenberg like groups studied in this paper allow us to give a more complete description of the square integrable holomorphic function spaces than was possible in [10][11][12] for the Hilbert-Schmidt groups.…”

Section: Discussionmentioning

confidence: 99%

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

Driver

Gordina

2009

Probab. Theory Relat. Fields

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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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“…In this paper we review our results in [7], [8], [9], [11] and show how they fit into a broader picture. In order to see the challenges this study presents, we first review what is known in finite dimensions and for the flat infinite-dimensional case.…”

Section: Motivation and Historymentioning

confidence: 90%

“…As we showed in [7], [8] and [9], there might be no nonconstant square integrable holomorphic functions. This question is related to another interesting problem which is new in infinite dimensions.…”

Section: New Features In the Infinite-dimensional Noncommutative Casementioning

confidence: 92%

Heat Kernel Analysis on Infinite Dimensional Groups

Gordina¹

2005

Infinite Dimensional Harmonic Analysis III

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Abstract. The paper gives an overview of our previous results concerning heat kernel measures in infinite dimensions. We give a history of the subject first, and then describe the construction of heat kernel measure for a class of infinite-dimensional groups. The main tool we use is the theory of stochastic differential equations in infinite dimensions. We provide examples of groups to which our results can be applied. The case of finite-dimensional matrix groups is included as a particular case. Motivation and history.In this paper we review our results in [7], [8], [9], [11] and show how they fit into a broader picture. In order to see the challenges this study presents, we first review what is known in finite dimensions and for the flat infinite-dimensional case. Our main results concern analogues of heat kernel (Gaussian) measures on infinitedimensional groups.1.1. Finite-dimensional noncommutative case. The initial motivation for our work was the following result for finite-dimensional Lie groups. It has a long history which we address later in this section. Let G be a finite-dimensional connected complex Lie group with a Lie algebra g. The Lie algebra is a complex Hilbert space with a norm | · |, and we denote by {ξ i } 2n i=1 an orthonormal basis of g as a real vector space. By identifying g with left-invariant vector fields at the identity e we can define the derivativeand the Laplacian ∆f = ∂ 2 i f. The heat kernel measure µ t has the heat kernel as its density with respect to a Haar measure dx. The heat kernel is the fundamental solution to the heat equation

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TAYLOR MAP ON GROUPS ASSOCIATED WITH A II₁-FACTOR

Cited by 9 publications

References 9 publications

The Taylor map on complex path groups

The Taylor map on complex path groups

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

Heat Kernel Analysis on Infinite Dimensional Groups

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TAYLOR MAP ON GROUPS ASSOCIATED WITH A II1-FACTOR

Cited by 9 publications

References 9 publications

The Taylor map on complex path groups

The Taylor map on complex path groups

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

Heat Kernel Analysis on Infinite Dimensional Groups

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Product

Resources

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TAYLOR MAP ON GROUPS ASSOCIATED WITH A II₁-FACTOR