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Proceedings of the American Mathematical Society

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Characterizations of $w^{\ast }$-homomorphisms and expectations


Author: T. W. Palmer
Journal: Proc. Amer. Math. Soc. 46 (1974), 265-272
MSC: Primary 46K99
DOI: https://doi.org/10.1090/S0002-9939-1974-0361804-1
MathSciNet review: 0361804
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Abstract: If $\mathfrak {A}$ and $\mathfrak {B}$ are $\ast$-algebras a map $\phi :\mathfrak {A} \to \mathfrak {B}$ is called a Schwarz map if it is linear and satisfies the Cauchy-Schwarz inequality $\phi {(a)^ \ast }\phi (a) \leq \phi (a ^\ast a)$ for all $a \in \mathfrak {A}$. Under mild restrictions on $\mathfrak {A}$ and $\mathfrak {B}$, $\ast$-homomorphisms and expectations are characterized in terms of Schwarz maps $\phi :\mathfrak {A} \to \mathfrak {B}$. The proofs are based on an elementary result on the multiplicative properties of Schwarz maps.


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Keywords: <IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img5.gif" ALT="$\ast$">-algebra, <IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\ast$">-homomorphism, Cauchy-Schwarz inequality, expectation, Banach <IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\ast$">-algebra, <!โ€“ MATH ${U^ \ast }$ โ€“> <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="${U^ \ast }$">-algebra, Jordan <IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$\ast$">-homomorphism
Article copyright: © Copyright 1974 American Mathematical Society