• Honam Mathematical Journal
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• v.28 no.4
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• pp.533-541
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• 2006

let E be a Finsler modules over $C^*$-algebras. A with norm-map $\rho$ and L(E) set of all A-linear bonded operators on E. We show that the canonical homomorphism ${\phi}:L(E){\rightarrow}L(E_I)$ sending each operator T to its restriction $T|E_I$ is injective if and only if I is an essential ideal in the underlying $C^*$-algebra A. We also show that $T{\in}L(E)$ is a bounded below if and only if ${\mid}{\mid}x{\mid}{\mid}={\mid}{\mid}{\rho}{\prime}(x){\mid}{\mid}$ is complete, where ${\rho}{\prime}(x)={\rho}(Tx)$ for all $x{\in}E$. Also, we give a necessary and sufficient condition for the equivalence of the norms generated by the norm map.

• Bulletin of the Korean Mathematical Society
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• v.25 no.2
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• pp.167-170
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• 1988

Versions of Tannaka duality in operator algebraic context have been obtained in [6], [8] etc. Suppose .sigma.is an automorphism of a von Neumann algebra M, on which there is an action .alpha. of a compact group G such that .sigma. vertical bar $M^{\alpha}$=id, where $M^{\tau}$is the fixed point algebra under the action .alpha.. Then it is shown that if there is an action .tau. of a group H which commutes with .alpha., and which is ergodic in the sense that the fixed point algebra $M^{\tau}$ is trivial, then there exists g.mem.G such that .sigma.=.alpha.(g). Recently Evans and Kishimoto ([4]) showed the versions of Tannaka duality in $C^{*}$-settings under some conditions.s.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.957-972
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• 2020

Let 𝓡 be a 2-torsion free unital ring containing a non-trivial idempotent. An additive map 𝛿 from 𝓡 into itself is called a Jordan derivable map at commutative zero point if 𝛿(AB + BA) = 𝛿(A)B + B𝛿(A) + A𝛿(B) + 𝛿(B)A for all A, B ∈ 𝓡 with AB = BA = 0. In this paper, we prove that, under some mild conditions, each Jordan derivable map at commutative zero point has the form 𝛿(A) = 𝜓(A) + CA for all A ∈ 𝓡, where 𝜓 is an additive Jordan derivation of 𝓡 and C is a central element of 𝓡. Then we generalize the result to the case of Jordan higher derivable maps at commutative zero point. These results are also applied to some operator algebras.

• Spatial Information Research
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• v.5 no.2
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• pp.147-160
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• 1997

"The "Map Algebra", beeing recognized as a viable theoretical framework for GIS (Geographica Infonnation System), models map layers as "operands" which are the basic unit of geo-processing, and a variety of GIS commands as "operators." In this paper, we attempt at lifting some limitations of map algebras proposed in GIS literature. First, we model map layer as "function" such that we may employ the notion of meta operator (or, higher-order funtion) available in the functional programming paradigm. This approach provides map algebraic language with "programmability" needed in GIS user language. Second, we extend the semantics of, and improve on the sytactic structure of map algebraic language. Mer the data model and language associated with map algebra are formalized, we proceed to design and implement a prototype of map algebraic processor. The parser of the language in our prototype plays the role of transforming the native and heterogeneous user language of current GISs into a canonical map algebraic language. The prototype, named "MapSee" is a proof-of-concept system for the ideas we propsed in this paper. We believe that the uniform interface based on the map algebraic language will make promising infrastructure to support "Internet GIS." This is because the uniform but powerful interface through the Web clients allow access to both geo-data and geo-processing resources distributed over the network.to both geo-data and geo-processing resources distributed over the network.