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Linear maps preserving AN-operators
Ramesh Golla,Hiroyuki Osaka 대한수학회 2020 대한수학회보 Vol.57 No.4
Let $H$ be a complex Hilbert space and $T : H\rightarrow H$ be a bounded linear operator. Then $T$ is said to be \textit{norm attaining} if there exists a unit vector $x_0\in H$ such that $\|Tx_0\|=\|T\|$. If for any closed subspace $M$ of $H$, the restriction $T|M : M \rightarrow H$ of $T$ to $M$ is norm attaining, then $T$ is called an \textit{absolutely norm attaining} operator or $\mathcal{AN}$-operator. In this note, we discuss linear maps on $\mathcal B(H)$, which preserve the class of absolutely norm attaining operators on $H$.
LINEAR MAPS PRESERVING 𝓐𝓝-OPERATORS
Golla, Ramesh,Osaka, Hiroyuki Korean Mathematical Society 2020 대한수학회보 Vol.57 No.4
Let H be a complex Hilbert space and T : H → H be a bounded linear operator. Then T is said to be norm attaining if there exists a unit vector x<sub>0</sub> ∈ H such that ║Tx<sub>0</sub>║ = ║T║. If for any closed subspace M of H, the restriction T|M : M → H of T to M is norm attaining, then T is called an absolutely norm attaining operator or 𝓐𝓝-operator. In this note, we discuss linear maps on B(H), which preserve the class of absolutely norm attaining operators on H.
On dualities of actions and inclusions
Lee, Hyun Ho,Osaka, Hiroyuki Elsevier 2019 Journal of functional analysis Vol.276 No.2
<P><B>Abstract</B></P> <P>Following the results known in the case of a finite abelian group action on <SUP> C ⁎ </SUP> -algebras we prove the following two theorems;<UL> <LI> an inclusion P ⊂ A of (Watatani) index-finite type has the Rokhlin property (is approximately representable) if and only if the dual inclusion is approximately representable (has the Rokhlin property). </LI> <LI> an inclusion P ⊂ A of (Watatani) index-finite type has the tracial Rokhlin property (is tracially approximately representable) if and only if the dual inclusion is tracially approximately representable (has the tracial Rokhlin property). </LI> </UL> Moreover, we provide an alternate proof of Phillips' theorem about the relations between tracial Rokhlin action and tracially approximate representable dual action using a new conceptual framework suggested by authors.</P>