# Topological Quantum Field Theory Seminar

### Ideals of nest algebras

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 $\left(a,b,c\right)\to abc$ mapping $A×{A}_{s}×A$ 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

1. 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.
2. Lina Oliveira, Weak*-closed Jordan ideals of nest algebras, Math. Nach., to appear.