## Algebraic isomorphisms and $\mathcal {J}$-subspace lattices

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- by Jiankui Li and Oreste Panaia PDF
- Proc. Amer. Math. Soc.
**133**(2005), 2577-2587 Request permission

## Abstract:

The class of $\mathcal {J}$-lattices was originally defined in the second author’s thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice $\mathcal {L}$ on a Banach space $X$ which is also a $\mathcal {J}$-lattice is called a $\mathcal {J}$-*subspace lattice*, abbreviated JSL. It is demonstrated that every single element of $Alg\mathcal {L}$ has rank at most one. It is also shown that $Alg\mathcal {L}$ has the strong finite rank decomposability property. Let $\mathcal {L}_1$ and $\mathcal {L}_2$ be subspace lattices that are also JSL’s on the Banach spaces $X_1$ and $X_2$, respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between $Alg\mathcal {L}_1$ and $Alg\mathcal {L}_2$ preserves rank. Finally we prove that every algebraic isomorphism between $Alg\mathcal {L}_1$ and $Alg\mathcal {L}_2$ is quasi-spatial.

## References

- K. R. Davidson, K. J. Harrison, and U. A. Mueller,
*Rank decomposability in incident spaces*, Linear Algebra Appl.**230**(1995), 3–19. MR**1355684**, DOI 10.1016/0024-3795(93)00351-Y - Frank Gilfeather and Robert L. Moore,
*Isomorphisms of certain CSL algebras*, J. Funct. Anal.**67**(1986), no. 2, 264–291. MR**845200**, DOI 10.1016/0022-1236(86)90039-X - A. Katavolos, M. S. Lambrou, and M. Papadakis,
*On some algebras diagonalized by $M$-bases of $l^2$*, Integral Equations Operator Theory**17**(1993), no. 1, 68–94. MR**1220574**, DOI 10.1007/BF01322547 - A. Katavolos, M. S. Lambrou, and W. E. Longstaff,
*Pentagon subspace lattices on Banach spaces*, J. Operator Theory**46**(2001), no. 2, 355–380. MR**1870412** - M. S. Lambrou,
*Approximants, commutants and double commutants in normed algebras*, J. London Math. Soc. (2)**25**(1982), no. 3, 499–512. MR**657507**, DOI 10.1112/jlms/s2-25.3.499 - M. S. Lambrou,
*Automatic continuity and implementation of homomorphisms*, (manuscript). - M. S. Lambrou and W. E. Longstaff,
*Non-reflexive pentagon subspace lattices*, Studia Math.**125**(1997), no. 2, 187–199. MR**1455633**, DOI 10.4064/sm-125-2-187-199 - M. S. Lambrou,
*On the rank of operators in reflexive algebras*, Linear Algebra Appl.**142**(1990), 211–235. MR**1077986**, DOI 10.1016/0024-3795(90)90268-H - Li Jiankui,
*Decomposability of certain reflexive algebras*, Houston J. Math.**23**(1997), no. 1, 121–126. MR**1688835** - W. E. Longstaff,
*Strongly reflexive lattices*, J. London Math. Soc. (2)**11**(1975), no. 4, 491–498. MR**394233**, DOI 10.1112/jlms/s2-11.4.491 - W. E. Longstaff, J. B. Nation, and Oreste Panaia,
*Abstract reflexive sublattices and completely distributive collapsibility*, Bull. Austral. Math. Soc.**58**(1998), no. 2, 245–260. MR**1642047**, DOI 10.1017/S0004972700032226 - W. E. Longstaff and Oreste Panaia,
*$\scr J$-subspace lattices and subspace $\rm M$-bases*, Studia Math.**139**(2000), no. 3, 197–212. MR**1762581** - W. E. Longstaff and Oreste Panaia,
*On the ranks of single elements of reflexive operator algebras*, Proc. Amer. Math. Soc.**125**(1997), no. 10, 2875–2882. MR**1402872**, DOI 10.1090/S0002-9939-97-03968-3 - W. E. Longstaff and Oreste Panaia,
*Single elements of matrix incidence algebras*, Linear Algebra Appl.**318**(2000), no. 1-3, 117–126. MR**1787228**, DOI 10.1016/S0024-3795(00)00165-8 - W. E. Longstaff and Oreste Panaia,
*Single elements of finite CSL algebras*, Proc. Amer. Math. Soc.**129**(2001), no. 4, 1021–1029. MR**1814141**, DOI 10.1090/S0002-9939-00-05714-2 - W. E. Longstaff,
*Operators of rank one in reflexive algebras*, Canadian J. Math.**28**(1976), no. 1, 19–23. MR**397435**, DOI 10.4153/CJM-1976-003-1 - Oreste Panaia,
*Quasi-spatiality of isomorphisms for certain classes of operator algebras*, Ph.D. dissertation, University of Western Australia (1995). - Oreste Panaia,
*Algebraic isomorphisms and finite distributive subspace lattices*, J. London Math. Soc. (2)**59**(1999), no. 3, 1033–1048. MR**1709095**, DOI 10.1112/S0024610799007450 - N. K. Spanoudakis,
*Operators in finite distributive subspace lattices. III*, Linear Algebra Appl.**262**(1997), 189–207. MR**1451775** - J. R. Ringrose,
*On some algebras of operators. II*, Proc. London Math. Soc. (3)**16**(1966), 385–402. MR**196516**, DOI 10.1112/plms/s3-16.1.385

## Additional Information

**Jiankui Li**- Affiliation: Department of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- Email: jli@math.uwaterloo.ca
**Oreste Panaia**- Affiliation: School of Mathematics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
- Email: oreste@maths.uwa.edu.au
- Received by editor(s): February 4, 2002
- Received by editor(s) in revised form: April 17, 2003
- Published electronically: April 15, 2005
- Communicated by: David R. Larson
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**133**(2005), 2577-2587 - MSC (2000): Primary 47L10
- DOI: https://doi.org/10.1090/S0002-9939-05-07581-7
- MathSciNet review: 2146201