Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a class of determinant preserving maps for finite von Neumann
  algebras   Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial
state $\tau $ and let $\Delta$ denote the associated Fuglede-Kadison
determinant. In this paper, we characterize all unital bijective maps $\phi$ on
the set of invertible positive elements in $\mathscr{R}$ which satisfy
$$\Delta(\phi(A)+\phi(B)) = \Delta(A+B).$$ We show that any such map originates
from a $\tau$-preserving Jordan $*$-automorphism of $\mathscr{R}$ (either
$*$-automorphism or $*$-anti-automorphism in the more restrictive case of
finite factors). In establishing the aforementioned result, we make crucial use
of the solutions to the equation $\Delta(A + B) = \Delta(A) + \Delta(B)$ in the
set of invertible positive operators in $\mathscr{R}$. To this end, we give a
new proof of the inequality $$\Delta(A+B) \ge \Delta(A) + \Delta(B),$$ using a
generalized version of the Hadamard determinant inequality and conclude that
equality holds for invertible $B$ if and only if $A$ is a nonnegative scalar
multiple of $B$.

12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn