When is the Isbell topology a group topology?

Conditions on a topological space X under which the space C ( X , R ) of continuous real-valued maps with the Isbell topology κ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in C κ ( X , R ) if and only if X is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in C κ ( X , R ) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if X is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.