×

A condition for a subspace of \({\mathcal B} (H)\) to be an ideal. (English) Zbl 0852.46021

Summary: Let \(H\) be a real or complex Hilbert space with \(\dim H> 1\). The subspace \({\mathcal A}\subset {\mathcal B}(H)\) is an ideal if and only if \(TA- AT^*\in {\mathcal A}\) for every \(T\in {\mathcal B}(H)\), \(A\in {\mathcal A}\). Every element of \({\mathcal B}(H)\) is the finite sum of \(TA- AT^*\) type operators.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
47L10 Algebras of operators on Banach spaces and other topological linear spaces

Keywords:

ideal
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brešar, M.; Zalar, B., On the structure of Jordan ∗-derivations, Colloq. Math., 63, 163-171 (1992) · Zbl 0786.46045
[2] Halmos, P. R., A Hilbert Space Problem Book (1967), Van Nostrand: Van Nostrand Princeton · Zbl 0144.38704
[3] Fong, C. K.; Miers, C. R.; Sourour, A. R., Lie and Jordan ideals of operators on Hilbert space, (Proc. Amer. Math. Soc., 84 (1982)), 516-520 · Zbl 0509.47035
[4] L. Molnár, The range of a Jordan ∗-derivation, Math. Japon.; L. Molnár, The range of a Jordan ∗-derivation, Math. Japon. · Zbl 0886.47019
[5] Molnár, L., On the range of a normal Jordan ∗-derivation, Comment. Math. Univ. Carolin., 35, 691-695 (1994) · Zbl 0821.47028
[6] Šemrl, P., On Jordan ∗-derivations and an application, Colloq. Math., 59, 241-251 (1990) · Zbl 0723.46044
[7] Šemrl, P., Ring derivations on standard operator algebras, J. Funct. Anal., 112, 318-324 (1993) · Zbl 0801.47024
[8] Šemrl, P., Quadratic and quasi-quadratic functionals, (Proc. Amer. Math. Soc., 119 (1993)), 1105-1113 · Zbl 0803.15024
[9] Šemrl, P., Jordan ∗-derivations of standard operator algebras, (Proc. Amer. Math. Soc., 120 (1994)), 515-519 · Zbl 0816.47040
[10] B. Zalar, Jordan ∗-derivation pairs and quadratic functionals on modules over ∗-rings, Preprint.; B. Zalar, Jordan ∗-derivation pairs and quadratic functionals on modules over ∗-rings, Preprint. · Zbl 0881.39019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.