Bibcode
Andruchow, Esteban; Varela, Alejandro
Referencia bibliográfica
eprint arXiv:math/0008144
Fecha de publicación:
8
2000
Número de citas
0
Número de citas referidas
0
Descripción
Let $B$ be a C$^*$-algebra and $X$ a C$^*$ Hilbert $B$-module. If $pin
B$ is a projection, denote by $S_p ={xin X : < x,x> =p}$, the
$p$-sphere of $X$. For $phi$ a state of $B$ with support $p$ in $B$ and
$xin S_p$, 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 $pin R_0$, 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 $$ S_p(R_0)={vin R_0:v^*v=p}subset R_0 $$ of
partial isometries with initial space $p$, regarded with the ultraweak
topology.