Modular Jordan algebras of self-adjoint operators
Teoretičeskaâ i matematičeskaâ fizika, Tome 53 (1982) no. 1, pp. 77-82
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An investigation is made into the connection between the type of a $JW$-algebra (i.e. , a weakly closed Jordan algebra of self-adjoint operators on a Hilbert space) and the type of the enveloping yon Neumann algebra. It is shown that every finite trace (faithful or normal) on a $JW$-algebra $A$ can be extended to a finite trace (faithful or normal, respectively) on the enveloping yon Neumann algebra $\mathfrak U(A)$. Using this result, it is shown that the $JW$ algebra $A$ is modular if and only if $\mathfrak U(A)$ is a finite yon
Neumann algebra. If $A$ is a reversible $JW$-factor, then it has the type $\operatorname{II}_1$ if and only if $\mathfrak U(A)$ has the type $\operatorname{II}_1$.
@article{TMF_1982_53_1_a7,
author = {Sh. A. Ayupov},
title = {Modular {Jordan} algebras of self-adjoint operators},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {77--82},
publisher = {mathdoc},
volume = {53},
number = {1},
year = {1982},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1982_53_1_a7/}
}
Sh. A. Ayupov. Modular Jordan algebras of self-adjoint operators. Teoretičeskaâ i matematičeskaâ fizika, Tome 53 (1982) no. 1, pp. 77-82. http://geodesic.mathdoc.fr/item/TMF_1982_53_1_a7/