Mediterr. J. Math. (2017) 14:184
published online August 14, 2017
Springer International Publishing AG 2017
Mati Abel, Lourdes Palacios and
Yuliana de Jes´us Z´arate-Rodr´ıguez
Abstract. 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.
Mathematics Subject Classiﬁcation. 46H05, 46H20.
Keywords. Topological algebras, Q-algebras, TQ-algebras, topological
spectrum, topological functional spectrum.
The Q-property was deﬁned by Kaplansky  in 1947 and was formulated
in the context of topological rings. It states that the set of quasi-invertible
elements in a topological ring (or topological algebra) is an open set. The
name of the property seems to be connected with the initial letter of the word
quasi. A topological algebra A is a Q-algebra if it satisﬁes the Q-property. Q-
algebras play an important role in topological algebra theory, sharing several
signiﬁcant properties of Banach algebras. We mention a few of them: in Q-
algebras every character is continuous, every maximal regular ideal is closed,
every element has a compact spectrum and so on.
As a very convenient generalization of invertibility, M. Abel , studied
the notion of topological quasi-invertibility of elements (topological invertibil-
ity,ifA is unital) and the advertibility of topological algebras (that is, the
class of topological algebras in which every topologically quasi-invertible el-
ement is quasi-invertible). The notion of TQ-algebras has been considered
only in the unital case (see [4,12]) or in the commutative case (see [1,8,9]).
The left and right topological quasi-invertibility of elements in topolog-
ical algebras, main properties of (not necessarily unital or commutative) left
and right TQ-algebras and topological spectrums of elements are studied in