## Rocky Mountain Journal of Mathematics

### Full-derivable points of $\mathcal {J}$-subspace lattice algebras

#### Abstract

Let $\mathcal{L}$ be a $\mathcal{J}$-subspace lattice on a complex Banach space $X$ and $\mbox{\rm Alg\,}{\mathcal L}$ the associated $\mathcal{J}$-subspace lattice algebra. We say that an operator $Z\in {\rm Alg\,}{\mathcal L}$ is a full-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a derivation and is a full-generalized-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself generalized derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)-A\delta(I)B=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a generalized derivation. In this paper, we prove that if $Z\in\mbox{\rm Alg\,}{\mathcal L}$ is an injective operator or an operator with dense range, then $Z$ is a full-derivable point as well as a full-generalized-derivable point of ${\rm Alg\,}{\mathcal L}$.

#### Article information

Source
Rocky Mountain J. Math., Volume 45, Number 1 (2015), 345-358.

Dates
First available in Project Euclid: 7 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1428412784

Digital Object Identifier
doi:10.1216/RMJ-2015-45-1-345

Mathematical Reviews number (MathSciNet)
MR3334215

Zentralblatt MATH identifier
1309.47041

#### Citation

Qi, Xiaofei; Hou, Jinchuan. Full-derivable points of $\mathcal {J}$-subspace lattice algebras. Rocky Mountain J. Math. 45 (2015), no. 1, 345--358. doi:10.1216/RMJ-2015-45-1-345. https://projecteuclid.org/euclid.rmjm/1428412784

#### References

• R.L. Crist, Local derivations on operator algebras, J. Funct. Anal. 135 (1996), 76–92.
• J.C. Hou and M.Y. Jiao, Additive derivable maps at zero point on nest algebras, Lin. Alg. Appl. 432 (2010), 2984–2994.
• J.C. Hou and X.F. Qi, Additive maps derivable at some points on $\mathcal{J}$-subspace lattice algebras, Lin. Alg. Appl. 429 (2008), 1851–1863.
• W. Jing, S.J. Lu and P.T. Li, Characterization of derivations on some operator algebras, Bull. Austr. Math. Soc. 66 (2002), 227–232.
• M.S. Lambrou, On the rank of operators in reflexive algebras, Lin. Alg. Appl. 142 (1990), 211–235.
• W.E. Longstaff, Strongly reflexive lattices, J. Lond. Math. Soc. 11 (1975), 491–498.
• ––––, Operators of rank one in reflexive algebras, Canad. J. Math. 28 (1976), 9–23.
• W.E. Longstaff, J.B. Nation and O. Panaia, Abstract reflexivity subspce lattices and completely distributive collapsibility, Bull. Austr. Math. Soc. 58 (1998), 245–260.
• W.E. Longstaff and O. Panaia, $\mathcal{J}$-subspace lattices and subspace $M$-bases, Stud. Math. 139 (2000), 197–211.
• S.J. Lu, F.Y. Lu, P.T. Li and Z. Dong, Non self-adjoint operator algebras, Science Press, Beijing, 2004.
• J. Zhu, All-derivable points of operator algebras, Lin. Alg. Appl. 427 (2007), 1–5.
• J. Zhu and C.P. Xiong, Bilocal derivations of standard operator algebras, Proc. Amer. Math. Soc. 125 (1997), 1367–1370.
• ––––, Generalized derivable mappings at zero point on nest algebras, Acta Math. Sinica 45 (2002), 783–788.
• ––––, Generalized derivable mappings at zero point on some reflexive operator algebras, Lin. Alg. Appl. 397 (2005), 367–379.
• ––––, Derivable mappings at unit operator on nest algebras, Lin. Alg. Appl. 422 (2007), 721–735.
• ––––, All-derivable points in continuous nest algebras, J. Math. Anal. Appl. 340 (2008), 845–853.