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 | References | Similar Articles | Additional Information
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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- Theodore W. Palmer, $^ *$-Representations of $U^ *$-algebras, Indiana Univ. Math. J. 20 (1971), no. 10, 929–933. MR 410396, DOI https://doi.org/10.1512/iumj.1971.20.20082 ---, $^\ast$-algebras (monograph in preparation).
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Additional Information
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
Article copyright:
© Copyright 1972
American Mathematical Society
