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In this paper left, right and two-sided Mackey $${ TQ}$$ -algebras are defined and main properties of these topological algebras are studied. Relations between Mackey $${ TQ}$$ -algebras with other topological algebras are given.

Topological spectral seminorm on topological algebras is studied. Main properties of topologically spectral algebras and topologically spectral subalgebras of topological algebras are described. Moreover, the descriptions of one-sided topological radicals of topological algebras are given.

A topological algebra A is called a Q-algebra if its set of quasi-invertible elements (of invertible elements, if A is unital) is an open set. This class of topological algebras has been extensively studied by several authors, but mostly in unital or commutative case. Properties of (not necessarily unital or commutative) TQ-algebras and, in the special case, of TQ-algebras with functional topological spectrum are given in the present paper.