On the homotopy type of the regular group of a real \(W^ *\)-algebra (Q1086475)

Trying to characterize a \(W^*\)-algebra factor \({\mathcal M}\) by the homotopical structure of its unitary group U(\({\mathcal M})\), \textit{H. Araki, M.-S. B. Smith} and \textit{L. Smith} [Commun. Math. Phys. 22, 71-88 (1971; Zbl 0211.440)] determined the fundamental group \(\pi_ 1(U,({\mathcal M}))\) [the non-factorial case has been considered in \textit{D. Handelman}, Quart. J. Math., Oxford, II. Ser. 29, 427-441 (1978; Zbl 0408.46057)]. Higher homotopy groups have been proved to be trivial if \({\mathcal M}\) is properly infinite by \textit{J. Brüning} and (INVALID INPUT)W. Willgerodt [Math. Ann. 220, 47-58 (1976; Zbl 0304.46040)], and to coincide with the corresponding homotopy groups of the stable unitary group if \({\mathcal M}\) is of type \(II_ 1\) in [the author, Math. Ann. 267, 271-277 (1984; Zbl 0525.46040)]. In the present paper we consider a purely real \(W^*\)-algebra \({\mathcal M}\), i.e. a weakly closed operator algebra on a real Hilbert space with trivial involution on the center. Then if \({\mathcal M}\) has no finite discrete part the following holds: \[ \pi_ k(U({\mathcal M}))=K_ 0({\mathcal M})\text{ if \(k=3\) (mod 4) and =0 otherwise.} \]