This paper is concerned
with modular annihilator A∗-algebras. Let A be an A∗-algebra, B a maximal
commutative ∗-subalgebra of A and XB the carrier space of B. We show that the
following statements are equivalent: (i) A is a modular annihilator algebra.
(ii) Every XB is discrete. (iii) Every B is a modular annihilator algebra.
(iv) The spectrum of every hermitian element of A has no nonzero limit
points.
Let A be an A∗-algebra which is a dense two-sided ideal of a B∗-algebra A,A∗∗
the second conjugate space of A and πA the canonical embedding of A into A∗∗. We
show that A is a modular annihilator algebra if and only if πA(A) is a two-sided ideal
of A∗∗ (with the Arens product). This generalizes a recent result by B. J. Tomiuk
and the author.
