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The set of double centralizers of A can be given a C*-algebra structure. This C*-algebra contains A as an essential ideal and can be identified as the multiplier algebra M ( A ). For instance, if A is the compact operators K ( H ) on a separable Hilbert space, then each x ∈ B ( H ) defines a double centralizer of A by simply multiplication ...
For example, there are four non-isomorphic groups of order 16 that are semidirect products of C 8 and C 2; in this case, C 8 is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups: the dihedral group of order 16; the quasidihedral group ...