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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Completely positive maps on $U^{\ast }$-algebras


Author: W. L. Paschke
Journal: Proc. Amer. Math. Soc. 34 (1972), 412-416
MSC: Primary 46K99
DOI: https://doi.org/10.1090/S0002-9939-1972-0300103-9
MathSciNet review: 0300103
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Abstract: Completely positive maps on ${U^ \ast }$-algebras with identity are characterized in terms of $^ \ast$-representations on Hilbert space. A result on restricted multiplicativity of such maps is established, from which it follows that completely positive maps which take unitaries to unitaries are $^ \ast$-homomorphisms. It is also shown that positive maps on commutative ${U^ \ast }$-algebras with identity are completely positive.


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Keywords: <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img9.gif" ALT="$^ \ast$">-algebra, <!– MATH ${B^ \ast }$ –> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${B^ \ast }$">-algebra, <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="${U^\ast }$">-algebra, completely positive, positive, <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img8.gif" ALT="$^\ast$">-representation, unitary
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