### Erratum/addendum to the paper: "Quasi *-algebras and generalized inductive limits of C*-algebras" (Studia Math. 202 (2011), 165-190)

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This paper may be viewed as having two aims. First, we continue our study of algebras of operators on a Hilbert space which have a contractive approximate identity, this time from a more Banach-algebraic point of view. Namely, we mainly investigate topics concerned with the ideal structure, and hereditary subalgebras (or HSA's, which are in some sense a generalization of ideals). Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which...

We generalize some technical results of Glicksberg to the realm of general operator algebras and use them to give a characterization of open and closed projections in terms of certain multiplier algebras. This generalizes a theorem of J. Wells characterizing an important class of ideals in uniform algebras. The difficult implication in our main theorem is that if a projection is open in an operator algebra, then the multiplier algebra of the associated hereditary subalgebra arises as the closure...

A generalized procedure for the construction of the inductive limit of a family of C*-algebras is proposed. The outcome is no more a C*-algebra but, under certain assumptions, a locally convex quasi *-algebra, named a C*-inductive quasi *-algebra. The properties of positive functionals and representations of C*-inductive quasi *-algebras are investigated, in close connection with the corresponding properties of positive functionals and representations of the C*-algebras that generate the structure....

Suppose $A$ is a separable unital $\mathcal{C}\left(X\right)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and ${K}_{1}$-injective ${C}^{*}$-algebra $\mathcal{D}$. We show that $A$ and $\mathcal{C}\left(X\right)\otimes \mathcal{D}$ are isomorphic as $\mathcal{C}\left(X\right)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.