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Wiener-Akutowicz-Helson-Lowdenslager-Masani factorization and Borel-Weil decomposition on measurable loop groups. (Factorisation de Wiener-Akutowicz-Helson-Lowdenslager-Masani et décomposition de Borel-Weil sur des groupes de lacets mesurés.) (French) Zbl 0904.60051

This paper considers the problem of factorizing loops with values in complex groups into the product of a holomorphic and a unitary part, generalizing the well-known results of Szegö for complex-valued functions. First the case of the torus is described in details. Given a continuous loop \(f\) with values in the group of invertible complex numbers, one has a unique factorization \(f= hu\) with \(u\) taking values of modulus one, and \(h\) the boundary value of a holomorphic map with no zero in the unit disk. It is shown that if \(f\) is taken at random with the distribution of a multiplicative Brownian loop, then the distribution of \(u\) is absolutely continuous with respect to the law of the Brownian bridge on the torus. For loops with values in complex groups, the logarithm is not applicable anymore, but one still has the Darboux differential \(z^{-1}dz\). Using this, a proof is sketched of the existence of a factorization \(f= hu\) of a continuous loop \(f\) with values in the complex Lie group \(\text{GL}_d(C)\), into the product of a loop \(u\) with values in the unitary group \(U(d)\), and a \(\text{GL}_d(C)\)-valued holomorphic function \(h\) on the unit disk. It is also stated that if \(f\) is taken with the law of the Brownian bridge on \(\text{GL}_d(C)\), then the law of \(u\) is absolutely continuous with respect to the law of the Brownian bridge on \(U(d)\), but the proof being too long for this short paper, it is deferred to a later publication.
Reviewer: Ph.Biane (Paris)

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60J65 Brownian motion
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps
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