Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fredholm operators in the Toeplitz Algebra $\mathcal{I}(QC)$   We will give a complete description of $\mathcal{I}$, the set of invertible
quasicontinuous functions on the unit circle. After doing this, we will then
classify the path-connected components of $\mathcal{I}$ and show that
$\mathcal{I}$ has uncountably many path-connected components. We will then use
the above classifications to characterize $\mathcal{F}$, the set of Fredholm
operators of the C$^*$-algebra generated by the Toeplitz operators $T_\phi$
with quasicontinuous symbols $\phi$. Then we will classify the path-connected
components of $\mathcal{F}$ and show that $\mathcal{F}$ also has uncountably
many path-connected components.

25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph