Homotopy of vector states

Abstract

Let $B$ be a C$^*$-algebra and $X$ a C$^*$ Hilbert $B$-module. If $p\in B$ is a projection, denote by $S_p =\{x\in X : < x,x> =p\}$, the $p$-sphere of $X$. For $\phi$ a state of $B$ with support $p$ in $B$ and $x\in S_p$, consider the state $\phi_x$ of $L_B(X)$ given by $\phi_x(t)= \phi(< x,t(x)>)$. In this paper we study certain sets associated to these states, and examine their topologic properties. As an application of these techniques, we prove that the space of states of the hyperfinite II$_1$ factor $R_0$, with support equivalent to a given projection $p\in R_0$, regarded with the norm topology (of the conjugate space of $R_0$), has trivial homotopy groups of all orders. The same holds for the space $$ S_p(R_0)=\{v\in R_0:v^*v=p\}\subset R_0 $$ of partial isometries with initial space $p$, regarded with the ultraweak topology.