| Title:
|
A characterization of reflexive spaces of operators (English) |
| Author:
|
Bračič, Janko |
| Author:
|
Oliveira, Lina |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
1 |
| Year:
|
2018 |
| Pages:
|
257-266 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that for a linear space of operators ${\mathcal M}\subseteq {\mathcal B}(\scr {H}_1,\scr {H}_2)$ the following assertions are equivalent. (i) ${\mathcal M} $ is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map $\Psi =(\psi _1,\psi _2)$ on a bilattice ${\rm Bil}({\mathcal M})$ of subspaces determined by ${\mathcal M}$ with $P\leq \psi _1(P,Q)$ and $Q\leq \psi _2(P,Q)$ for any pair $(P,Q)\in {\rm Bil}({\mathcal M})$, and such that an operator $T\in {\mathcal B}(\scr {H}_1,\scr {H}_2)$ lies in ${\mathcal M}$ if and only if $\psi _2(P,Q)T\psi _1(P,Q)=0$ for all $(P,Q)\in {\rm Bil}( {\mathcal M})$. This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces. (English) |
| Keyword:
|
reflexive space of operators |
| Keyword:
|
order-preserving map |
| MSC:
|
47A15 |
| idZBL:
|
Zbl 06861579 |
| idMR:
|
MR3783597 |
| DOI:
|
10.21136/CMJ.2017.0456-16 |
| . |
| Date available:
|
2018-03-19T10:30:35Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147133 |
| . |
| Reference:
|
[1] Han, D. G.: On $\scr A$-submodules for reflexive operator algebras.Proc. Am. Math. Soc. 104 (1988), 1067-1070. Zbl 0694.47031, MR 0969048, 10.2307/2047592 |
| Reference:
|
[2] Erdos, J. A.: Reflexivity for subspace maps and linear spaces of operators.Proc. Lond. Math. Soc., III Ser. 52 (1986), 582-600. Zbl 0609.47053, MR 0833651, 10.1112/plms/s3-52.3.582 |
| Reference:
|
[3] Erdos, J. A., Power, S. C.: Weakly closed ideals of nest algebras.J. Oper. Theory 7 (1982), 219-235. Zbl 0523.47027, MR 0658610 |
| Reference:
|
[4] Hadwin, D.: A general view of reflexivity.Trans. Am. Math. Soc. 344 (1994), 325-360. Zbl 0802.46010, MR 1239639, 10.1090/S0002-9947-1994-1239639-4 |
| Reference:
|
[5] Halmos, P. R.: Reflexive lattices of subspaces.J. Lond. Math. Soc., II. Ser. 4 (1971), 257-263. Zbl 0231.47003, MR 0288612, 10.1112/jlms/s2-4.2.257 |
| Reference:
|
[6] Kliś-Garlicka, K.: Reflexivity of bilattices.Czech. Math. J. 63 (2013), 995-1000. Zbl 1313.47024, MR 3165510, 10.1007/s10587-013-0067-4 |
| Reference:
|
[7] Kliś-Garlicka, K.: Hyperreflexivity of bilattices.Czech. Math. J. 66 (2016), 119-125. Zbl 06587878, MR 3483227, 10.1007/s10587-016-0244-3 |
| Reference:
|
[8] Li, P., Li, F.: Jordan modules and Jordan ideals of reflexive algebras.Integral Equations Oper. Theory 74 (2012), 123-136. Zbl 1286.47046, MR 2969043, 10.1007/s00020-012-1982-8 |
| Reference:
|
[9] Loginov, A. I., Sul'man, V. S.: Hereditary and intermediate reflexivity of $W\sp*$-algebras.Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 1260-1273 Russian. Zbl 0327.46073, MR 0405124 |
| Reference:
|
[10] Shulman, V., Turowska, L.: Operator synthesis I. Synthetic sets, bilattices and tensor algebras.J. Funct. Anal. 209 (2004), 293-331. Zbl 1071.47066, MR 2044225, 10.1016/S0022-1236(03)00270-2 |
| . |
