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Friday, July 26, 2019 - 15:30 to 16:30
Mathematics Conference Room
Amit kumar
Isometries of absolute order unit spaces

Abstract: In this talk, we shall prove that a linear map between two absolute order unit spaces is an \emph{absolute value preserving} if and only if it is orthogonality preserving. With the help of this result, we proved that a unital, bijective linear map between two absolute order unit spaces is an isometry if and only if it is \emph{absolute value preserving} which leads to provide a simple proof of well known result that every unital Jordan isomorphism between two $JB$-algebras is an isometry and hence we deduce that unital, bijective \emph{absolute value preserving} maps between two $JB$-algebras are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces in the context of matrix ordered spaces and present a matricial version of these results. We prove that a unital, bijective $\ast$-linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is a \emph{complete absolute value preserving}. From here, we deduce a known result that on (unital) C$^*$-algebras such maps are precisely C$^*$-algebra isomorphisms. We give a simple, order-theoretic proof using a trick which is apparently new.

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