I am interested in known topologies that one can define on operators on a vector lattice or a topological vector lattice? For example, a known topology on L_b(E), the space of all order bounded operators on an order complete vector lattice E using order convergence of E? Or a topology on a L_r(E), the space of all regular operators on a topological vector lattice E using order and topological structures of E?
This is a good question.
The beginning of answer to this question can be found in
J. de D. Marques, On vectorial inner product spaces, Czechoslovak Math. J. 50 (2000), no. 3, 539-550:
http://dml.cz/bitstream/handle/10338.dmlcz/127591/CzechMathJ_50-2000-3_8.pdf
See Section 2.6 (regulart vectorial norms), stating on page 541, where Banach lattices are considered with the topology induced by the vectorial norm $p$. Completeness is considered in Def. 3.1.1, page 543 and Theorem 4.1.1 (norm complete Yosida algebra of all bound real-valued mapping (page 544).
More to the point, see Boolean algebras of projections considered in
P.G. Dodds, Boolean algebras in locally convex spaces:
http://maths-proceedings.anu.edu.au/CMAProcVol5/CMAProcVol5-Dodds.pdf
See Theorem 2.3, page 75, where an integration map on $\mathcal{B}$, a Bade complete Boolean algebra of projections, induces on the closed algebra generated by $\mathcal{B}$ the structure of an order complete, topologically complete convex vector lattice with a topology defined by order continuous lattice semi-norms.
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