engleski

The closure of two-sided multiplications on C*- algebras and phantom line bundles

For a C∗ -algebra A, we consider the problem when the set TM0(A) of all two-sided multiplications x↦axb(a, b∈A) on A is norm closed, as a subset of B(A)⁠. We first show that TM0(A) is norm closed for all prime C∗-algebras A⁠. On the other hand, if A≅Γ0(E) is an n-homogeneous C∗-algebra, where E is the canonical Mn-bundle over the primitive spectrum X of A⁠, we show that TM0(A) fails to be norm closed if and only if there exists a σ-compact open subset U of X and a phantom complex line subbundle L of E over U (i.e., L is not globally trivial, but is trivial on all compact subsets of U⁠). This phenomenon occurs whenever n≥2 and X is a CW-complex (or a topological manifold) of dimension 3≤d<∞⁠.

C*-algebra ; homogeneous ; two-sided multiplication ; elementary operator ; complex line bundle

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