Homotopy on State Orbits
by
Alejandro Varela
Universidad Nac. de Gral. Sarmiento
Coauthors: Esteban Andruchow (Universidad Nac. de Gral. Sarmiento)

Let j be a faithful, normal state on a von Neumann algebra M. Denote by U_M the unitary group of M, and by M^j the centralizer of j. Let U_j be the unitary orbit of j, i.e.

Uj={ j o Ad(u) : u in UM}

where Ad(u)(x)=uxu^*. In previous papers we introduced a homogeneous and reductive structure for U_j, by means of the natural identification U_j =~ U_M / U_Mj. We regard U_j with the quotient topology induced by the usual norm of M. With this topology it is a real analytic manifold. We prove that these orbits are always simply connected.

We also study another natural topology in the set U_M / U_M^j. Namely, let E_j be the unique j-invariant conditional expectation E_j:M --> M^j. This gives rise to a natural pre-Hilbert C^*-module structure for M with M^j-valued inner product given by < x, y > =E_j(x^*y) and norm ||x||_Ej=||E_j(x^*x)||^1/2.

We also present different models for U_j, inside the projections of the basic extension of E_j, and inside the interior tensor product M \otimes_M^j M of M regarded as a pre-Hilbert module M.

Date received: April 29, 1999