Filter games and pathological subgroups of a countable product of lines

To each filter F on ω, a certain linear subalgebra A(F) of Rω, the countable product of lines, is assigned. This algebra is shown to have many interesting topological properties, depending on the properties of the filter F. For example, if F is a free ultrafilter, then A(F) is a Baire subalgebra of Rω for which the game OF introduced by Tkachenko is undetermined (this resolves a problem of Hernández, Robbie and Tkachenko); and if F1 and F2 are two free filters on ω that are not near coherent (such filters exist under Martin's Axiom), then A (F1) and A(F2) are two o-bounded and OF-undetermined subalgebras of Rω whose product A(F1) × A(F2) is OF-determined and not o-bounded (this resolves a problem of Tkachenko). It is also shown that the statement that the product of two o-bounded subrings of Rω is o-bounded is equivalent to the set-theoretic principle NCF (Near Coherence of Filters); this suggests that Tkachenko's question on the productivity of the class of o-bounded topological groups may be undecidable in ZFC.