Complex Banach spaces are naturally endowed with an algebraic
structure, other than that of a vector space. The
holomorphic structure of the open unit ball in a complex
Banach space A leads to the existence of a closed subspace

${A}_{s}$ of

$A$, known as the symmetric part of

$A$, and of a partial triple product

$(a,b,c)\to abc$ mapping

$A\times {A}_{s}\times A$ to

$A$. The existence of a Jordan triple identity satisfyied by this algebraic
structure
relates any complex Banach space to the Banach Jordan triple systems
important in infinite-dimensional holomorphy.

A nest algebra, which is a primary example of a non-self-adjoint algebra of
operators, is also an
interesting case of a complex Banach space whose symmetric part is a proper subspace.

The ideals
of nest algebras related to its associative
multiplication have been extensively investigated, and
whilst it is clear
that ideals in the associative sense provide
examples of ideals in the partial triple sense, the converse
assertion remains in general an open problem.
It is the aim of this talk to show that, in a large class
of nest algebras, the weak*-closed ideals in the partial triple
sense are also
weak*-closed ideals in the associative algebra sense.

A brief overview of how the partial triple produtct arises from the
holomorphic structure of the open unit ball of the nest algebra will also be given.

#### References

- Jonathan Arazy,
*An application of infinite dimensional
holomorphy to the geometry of Banach space*, Geometrical aspects
of functional analysis, Lecture Notes in Mathematics,
Vol. 1267, Springer-Verlag, Berlin/Heidelberg/New York, 1987.
- Lina Oliveira,
*Weak*-closed Jordan ideals
of nest algebras*, Math. Nach., to appear.