Duality and Normal Parts of Operator Modules

For an operator bimodule $X$ over von Neumann algebras A\subseteq\bh and B\subseteq\bk, the space of all completely bounded $A,B$-bimodule maps from $X$ into \bkh, is the bimodule dual of $X$. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To $X$ a normal operator bimodule \nor{X} is associated so that completely bounded $A,B$-bimodule maps from $X$ into normal operator bimodules factorize uniquely through \nor{X}. A construction of \nor{X} in terms of biduals of $X$, $A$ and $B$ is presented. Various operator bimodule structures are considered on a Banach bimodule admitting a normal such structure.Comment: The first version of the paper has been split into two parts, corrected and a few results added. This is the first par

Variance of operators and derivations

The variance of a bounded linear operator $a$ on a Hilbert space $H$ at a unit vector $h$ is defined by $D_h(a)=\|ah\|^2-||^2$. We show that two operators $a$ and $b$ have the same variance at all vectors $h\in H$ if and only if there exist scalars $\sigma,\lambda$ with $|\sigma|=1$ such that $b=\sigma a+\lambda1$ or $a$ is normal and $b=\sigma a^*+\lambda1$. Further, if $a$ is normal, then the inequality $D_h(b)\leq\kappa D_h(a)$ holds for some constant $\kappa$ and all unit vectors $h$ if and only if $b=f(a)$ for a Lipschitz function $f$ on the spectrum of $a$. Variants of these results for C$^*$-algebras are also proved. We also study the related, but more restrictive inequalities $\|bx-xb\|\leq \|ax-xa\|$ supposed to hold for all $x\in B(H)$ or for all $x\in B(H^n)$ and all positive integers $n$. We consider the connection between such inequalities and the range inclusion $d_b(B(H))\subseteq d_a(B(H))$, where $d_a$ and $d_b$ are the derivations on $B(H)$ induced by $a$ and $b$. If $a$ is subnormal, we study these conditions in particular in the case when $b$ is of the form $b=f(a)$ for a function $f$.Comment: 31 pages, to appear in JMAA. The paper has been reorganized and the proofs of a few results corrected. The statement of the former Theorem 5.8 (now Corollary 5.4) has been change

Fixed points of normal completely positive maps on B(H)

Given a sequence of bounded operators $a_j$ on a Hilbert space $H$ with $\sum a_j^*a_j=1=\sum a_ja_j^*$, we study the map $\Psi$ defined on $B(H)$ by $\Psi(x)=\sum a_j^*xa_j$ and its restriction $\Phi$ to the Hilbert-Schmidt class $C^2(H)$. In the case when the sum $\sum a_j^*a_j$ is norm-convergent we show in particular that the operator $\Phi-1$ is not invertible if and only if the C$^*$-algebra $A$ generated by $(a_j)$ has an amenable trace. This is used to show that $\Psi$ may have fixed points in $B(H)$ which are not in the commutant $A'$ of $A$ even in the case when the weak* closure of $A$ is injective. However, if $A$ is abelian, then all fixed points of $\Psi$ are in $A'$ even if the operators $a_j$ are not positive.Comment: 17 page