arXiv:0907.4645v1 [math.OA] 27 Jul 2009 The type III 1 factor generated by regular representations of the infinite dimensional nilpotent group B Z 0 Ivan Dynov Department of Mathematics and Statistics, York University N536 Ross Building, 4700 Keele Street, Toronto, ON M3J 1P3 Canada Max-Planck-Institut f¨ ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany, E-mail: dynov@mpim-bonn.mpg.de and Alexandre Kosyak ∗ Max-Planck-Institut f¨ ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany Institute of Mathematics, Ukrainian National Academy of Sciences, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine, E-mail: kosyak01@yahoo.com, kosyak@imath.kiev.ua tel.: 38044 2346153 (office), 38044 5656758 (home), fax: 38044 2352010 Preprint submitted to