On centralizers of standard operator algebras with involution
The purpose of this paper is to prove the following result. Let <img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1106" width=18 height=25 border=0 id="_x0000_i1106"> be a complex Hilbert space, let <img src="http:/fbpe/img/cubo/v15n3/art05...
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| Autores principales: | , , |
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| Lenguaje: | English |
| Publicado: |
Universidad de La Frontera. Departamento de Matemática y Estadística.
2013
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| Materias: | |
| Acceso en línea: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462013000300005 |
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| Sumario: | The purpose of this paper is to prove the following result. Let <img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1106" width=18 height=25 border=0 id="_x0000_i1106"> be a complex Hilbert space, let <img src="http:/fbpe/img/cubo/v15n3/art05-fig2.jpg" name="_x0000_i1105" width=19 height=24 border=0 id="_x0000_i1105"> (<img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1104" width=18 height=25 border=0 id="_x0000_i1104">) be the algebra of all bounded linear operators on <img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1103" width=18 height=25 border=0 id="_x0000_i1103"> and let <img src="http:/fbpe/img/cubo/v15n3/art05-fig3.jpg" name="_x0000_i1102" width=22 height=24 border=0 id="_x0000_i1102"> (<img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1101" width=18 height=25 border=0 id="_x0000_i1101">) <img src="http:/fbpe/img/cubo/v15n3/art05-fig6.jpg" name="_x0000_i1100" width=17 height=18 border=0 id="_x0000_i1100"><img src="http:/fbpe/img/cubo/v15n3/art05-fig2.jpg" name="_x0000_i1099" width=19 height=24 border=0 id="_x0000_i1099">(<img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1098" width=18 height=25 border=0 id="_x0000_i1098">) be a standard operator algebra, which is closed under the adjoint operation. Let <img src="http:/fbpe/img/cubo/v15n3/art05-fig4.jpg" name="_x0000_i1097" width=15 height=21 border=0 id="_x0000_i1097">: <img src="http:/fbpe/img/cubo/v15n3/art05-fig3.jpg" name="_x0000_i1096" width=22 height=24 border=0 id="_x0000_i1096"> (<img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1095" width=18 height=25 border=0 id="_x0000_i1095">) <img src="http:/fbpe/img/cubo/v15n3/art03-fig1.jpg" name="_x0000_i1094" width=25 height=16 border=0 id="_x0000_i1094"> <img src="http:/fbpe/img/cubo/v15n3/art05-fig2.jpg" name="_x0000_i1093" width=19 height=24 border=0 id="_x0000_i1093">(<img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1092" width=18 height=25 border=0 id="_x0000_i1092">) be a linear mapping satisfying the relation 2<img src="http:/fbpe/img/cubo/v15n3/art05-fig4.jpg" name="_x0000_i1091" width=15 height=21 border=0 id="_x0000_i1091">(<img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1090" width=18 height=21 border=0 id="_x0000_i1090"><img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1089" width=18 height=21 border=0 id="_x0000_i1089">*<img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1088" width=18 height=21 border=0 id="_x0000_i1088">) = <img src="http:/fbpe/img/cubo/v15n3/art05-fig4.jpg" name="_x0000_i1087" width=15 height=21 border=0 id="_x0000_i1087">(<img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1086" width=18 height=21 border=0 id="_x0000_i1086">)<img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1085" width=18 height=21 border=0 id="_x0000_i1085">*<img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1084" width=18 height=21 border=0 id="_x0000_i1084"> + <img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1083" width=18 height=21 border=0 id="_x0000_i1083"><img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1082" width=18 height=21 border=0 id="_x0000_i1082">*<img src="http:/fbpe/img/cubo/v15n3/art05-fig4.jpg" name="_x0000_i1081" width=15 height=21 border=0 id="_x0000_i1081">(<img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1080" width=18 height=21 border=0 id="_x0000_i1080">) for all <img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1079" width=18 height=21 border=0 id="_x0000_i1079"><img src="http:/fbpe/img/cubo/v15n3/art05-fig7.jpg" name="_x0000_i1078" width=17 height=17 border=0 id="_x0000_i1078"><img src="http:/fbpe/img/cubo/v15n3/art05-fig3.jpg" name="_x0000_i1077" width=22 height=24 border=0 id="_x0000_i1077">(<img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1076" width=18 height=25 border=0 id="_x0000_i1076">). In this case <img src="http:/fbpe/img/cubo/v15n3/art05-fig4.jpg" name="_x0000_i1075" width=15 height=21 border=0 id="_x0000_i1075">is of the form <img src="http:/fbpe/img/cubo/v15n3/art05-fig4.jpg" name="_x0000_i1074" width=15 height=21 border=0 id="_x0000_i1074">(<img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1073" width=18 height=21 border=0 id="_x0000_i1073">) = λ<img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1072" width=18 height=21 border=0 id="_x0000_i1072"> for all <img src="http:/fbpe/img/cubo/v15n3/art05-fig5.jpg" name="_x0000_i1071" width=18 height=21 border=0 id="_x0000_i1071"><img src="http:/fbpe/img/cubo/v15n3/art05-fig7.jpg" name="_x0000_i1070" width=17 height=17 border=0 id="_x0000_i1070"><img src="http:/fbpe/img/cubo/v15n3/art05-fig3.jpg" name="_x0000_i1069" width=22 height=24 border=0 id="_x0000_i1069">(<img src="http:/fbpe/img/cubo/v15n3/art05-fig1.jpg" name="_x0000_i1068" width=18 height=25 border=0 id="_x0000_i1068">), where λ is some fixed complex number. |
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