# A note on zero Lie product determined nest algebras as Banach algebras

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran.

2 Department of Mathematics, Payam Noor University of Technology, P.O. Box 19395-3697, Tehran, Iran.

10.22072/wala.2020.130358.1293

Abstract

A Banach algebra $\A$ is said to be zero Lie product determined Banach algebra if for every continuous bilinear functional $\phi:\A \times \A\rightarrow \mathbb{C}$ the following holds: if $\phi(a,b)=0$ whenever $ab=ba$, then there exists some $\tau \in \A^*$ such that $\phi(a,b)=\tau(ab-ba)$ for all $a,b\in \A$. We show that any finite nest algebra over a complex Hilbert space is a zero Lie product determined Banach algebra.

Keywords

#### References

[1] J. Alaminos, M. Bre$check{textrm{s}}$ar, J. Extremera, $check{textrm{S}}$. $check{textrm{S}}$penko and A.R. Villena, Commutators and square-zero elements in Banach algebras, Q. J. Math., 67 (2016), 1-13.
[2] J. Alaminos, M. Bre$check{textrm{s}}$ar, J. Extremera and A.R. Villena, Maps preserving zero products, Studia Math., 193 (2009), 131-159.
[3] J. Alaminos, M. Bre$check{textrm{s}}$ar, J. Extremera and A.R. Villena, newblock Zero Lie product determined Banach algebras, II, J. Math. Anal. Appl., 474 (2019), 1498-1511.
[4] J. Alaminos, M. Bre$check{textrm{s}}$ar, J. Extremera and A.R. Villena, Zero Lie product determined Banach algebras,  Studia Math., 239 (2017), 189-199.
[5] M. Bre$check{textrm{s}}$ar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. R. Soc. Edinb., Sect. A, Math., 137 (2007), 9-21.
[6] M. Bre$check{textrm{s}}$ar, Finite dimensional zero product determined algebras are generated by idempotents, Expo. Math., 34 (2016), 130-143.
[7] M. Bre$check{textrm{s}}$ar, Functional identities and zero Lie product determined Banach algebras, Q. J. Math., 71  (2020), 649-665.
[8] M. Bre$check{textrm{s}}$ar, M. Gra$check{textrm{s}}$i$check{textrm{c}}$ and J. Sanchez, Zero product determined matrix algebras, Linear Algebra Appl., 430 (2009), 1486-1498.
[9] M. Bre$check{textrm{s}}$ar, Multiplication algebra and maps determined by zero products, Linear Multilinear Algebra, 60 (2012), 763-768.
[10] M. Bre$check{textrm{s}}$ar and P. $check{textrm{S}}$emrl, On bilinear maps on matrices with applications to commutativity preservers, J. Algebra, 301 (2006), 803-837.
[11] H.G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, New Series, 24, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2000.
[12] K.R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics, 191, Longman, London, 1988.
[13] B.E. Forrest and L.W. Marcoux, Derivations of triangular Bnach algebras, Indiana Univ. Math. J., 45  (1996), 441-462.
[14]] B.E. Forrest and L.W. Marcoux, Weak amenability of triangular Banach algebras, Trans. Am. Math. Soc., 345 (2002), 1435-1452.
[15] H. Ghahramani, Zero product determined some nest algebras, Linear Algebra Appl., 438 (2013), 303-314.
[16] H. Ghahramani, Zero product determined triangular algebras, Linear Multilinear Algebra, 61 (2013), 741-757.
[17] M. Gra$check{textrm{s}}$i$check{textrm{c}}$, Zero product determined classical Lie algebras, Linear Multilinear Algebra, 58 (2010), 1007-1022.
[18] C. Pearcy and D. Topping, Sum of small numbers of idempotent, Mich. Math. J., 14 (1967), 453-465.
[19] D. Wang, X. Yu and Z. Chen, newblock A class of zero product determined Lie algebras, J. Algebra, 331 (2011), 145-151.