An extension of compact operators by compact operators with no nontrivial multipliers

摘要

We construct a nonhomogeneous, separably represented, type I and approximately finite dimensional C*-algebra such that its multiplier algebra is equal to its unitization. This algebra is an essential extension of the algebra K(l(2) (c)) of compact operators on a nonseparable Hilbert space by the algebra K(l(2)) of compact operators on a separable Hilbert space, where c denotes the cardinality of continuum. Although both K(l(2) (c)) and K (l(2)) are stable, our algebra is not. This sheds light on the permanence properties of the stability in the nonseparable setting. Namely, unlike in the separable case, an extension of a stable nonseparable C*-algebra by K (l(2)) does not have to be stable. Our construction can be considered as a noncommutative version of Mrowka's Psi-space; a space whose one point compactification is equal to its Cech-Stone compactification and is induced by a special uncountable family of almost disjoint subsets of N.