oa.operator algebras – Schaten p norm of block matrices

Let $A=Doplus 0$ be a diagonal Hermitian matrix and $B$ is an invertible Hermitian matrix with $(1,1)$ block being $B_{11}$ and $B_{11}$ and $D$ have the same dimensions. Then is it true that if $(1+|t|^p)^{q/p}=|A+tB|_q^q$ for all $tinmathbb R,$ then $B_{12}=0$ where $B_{12}$ is in the $(1,2)$ block of $B$?

$|.|_q$ denotes the Schatten-q norm and $1<p,qneq 2<infty$

I somehow think that the answer is yes but cannot really prove it.

oa.operator algebras – Lower bounds in the space of compact operators

Let $H$ be a separable Hilbert space, and $K(H)$ the corresponding space of compact operators. Consider the “unit sphere” $S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$. Is it true that, given any pair of operators $T_1,T_2in S$, there exists another operator $Tin S$ such that $Tleq T_1,T_2$?.

Algebras oa.operator – Equivalence of the $ sigma-weak topology with respect to another topology

Let $ mathcal H $ to be a Hilbert space. Define a topology $ tau_1 $ sure $ B ( mathcal H) $ by the seminorms family $ x mapsto | Tr (xa) |, $ $ a in L ^ 1 (B ( mathcal H)). $ Right here $ B ( mathcal H) $ refers to the set of all linear maps delineated on $ mathcal H $ and $ L ^ 1 (B ( mathcal H)) $ refers to the operators of the trace class.
Again define the $ sigma $Topology -WOT $ tau_2 $ sure $ B ( mathcal H) $ by removing the weak operator topology from $ B ( mathcal H otimes ell_2) $ at $ B ( mathcal H) $ via the map $ x mapsto x otimes 1. $ How to show that $ tau_1 = tau_2 $? In many books and lecture notes in von Neumann's algebras, they have just mentioned that this is true. But I could not find any solid proof.