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 | References | Similar Articles | Additional Information
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