Fourier transform of supermeasures. (English) Zbl 1180.46058

Panyunin, N. M.

Russ. J. Math. Phys. 14, No. 4, 501-504 (2007).

Summary: A Smolyanov-Shavgulidze model of an infinite-dimensional superspace
\[ H_{\Lambda } = \Lambda _{0}\widehat {\otimes }H_{0} \oplus \Lambda _{1}\widehat {\otimes }H_{1} \]
corresponding to a Hilbert space \(H = H_{0} \oplus H_{1}\) over a Hilbert superalgebra \(\Lambda = \Lambda _{0} \oplus \Lambda _{1}\) is considered. Its relation to the Khrennikov superspace is discussed. Moreover, \(H_{\Lambda }\) is equipped with the structure of Hilbert superspace with an inner superproduct \((\cdot ,\cdot )_{\Lambda }\). The supermeasures are defined as \(\Lambda ^{\mathbb C}\widehat {\otimes }(\land H_{1})\)-valued measures on \(H_{0}(\Lambda ^{\mathbb C}\) stands for the complexification of \(\Lambda )\). The definition of the Fourier supertransform is similar to the ordinary one, \(\tilde {\mu }(y) = \smallint _{H_\Lambda }e^{i(y,z)_\Lambda }\mu (dz)\). It turns out that the values of the Fourier supertransform of a supermeasure on the subspace \(\overline H_{\Lambda } = H_{0} \oplus \Lambda _{1}\widehat {\otimes }H_{1}\) can be obtained by applying a certain operator to the values of the classical Fourier transform. The main result is the theorem claiming that the Fourier supertransform of supermeasures is isometric on superspaces with zero even part \((H_{0} = \{0\})\). As a corollary, we note that the operator of Fourier supertransform is injective. We also give necessary and sufficient conditions for the countable additivity of cylindrical supermeasures in terms of continuity of their Fourier supertransforms (an analogue of the Minlos-Sazonov theorem).

MSC:

46S60	Functional analysis on superspaces (supermanifolds) or graded spaces
58C50	Analysis on supermanifolds or graded manifolds

Keywords:

Smolyanov-Shavgulidze model; Hilbert superspace; supermeasures; Fourier supertransform

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