Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The closure of two-sided multiplications on C*-algebras and phantom line
  bundles   For a C*-algebra A we consider the problem of when the set $TM_0(A)$ of all
two-sided multiplications $x \mapsto axb$ ($a,b \in A$) on A is norm closed, as
a subset of B(A). We first show that $TM_0(A)$ is norm closed for all prime
C*-algebras A. On the other hand, if $A\cong \Gamma_0(E )$ is an n-homogeneous
C*-algebra, where E is the canonical $\mathbb{M}_n $-bundle over the primitive
spectrum X of A, we show that $TM_0(A)$ fails to be norm closed if and only if
there exists a $\sigma$-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 $n \geq 2$ and X is a
CW-complex (or a topological manifold) of dimension $3 \leq d<\infty$.

13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con