K. K. Kudaybergenov

Let $M$ be a type I von Neumann algebra with a center $Z,$ and a faithful normal semi-finite trace $\tau.$ Consider the algebra $L(M, \tau)$ of all $\tau$-measurable operators with respect to $M$ and let $S_0(M, \tau)$ be the subalgebra of $\tau$-compact operators in $L(M, \tau).$ We prove that any $Z$-linear involutive automorphisms of $S_0(M, \tau)$ is inner.

The paper is devoted to studying $\nabla$-Fredholm operators in Banach--Kantorovich spaces over a ring of measurable functions. We show that a bounded linear operator acting in Banach--Kantorovich space is $\nabla$-Fredholm if and only if it can be represented as a sum of an invertible operator and a cyclically compact operator.