075-02-2022-882
Let \mathcal{M} be a semifinite von Neumann algebra on a Hilbert space \mathcal{H} equipped with a faithful normal semifinite trace \tau, S(\mathcal{M},\tau) be the {}^*-algebra of all \tau-measurable operators. Let S_0(\mathcal{M},\tau) be the {}^*-algebra of all \tau-compact operators and T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I be the {}^*-algebra of all operators X=A+\lambda I
with A\in S_0(\mathcal{M},\tau) and \lambda \in \mathbb{C}. It is proved that every operator of T(\mathcal{M},\tau) that is left-invertible in T(\mathcal{M},\tau) is in fact invertible in T(\mathcal{M},\tau).
It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in \mathcal{B} (\mathcal{H}).
For the singular value function \mu(t; Q) of Q=Q^2\in S(\mathcal{M},\tau), the inclusion \mu(t; Q)\in \{0\}\bigcup
[1, +\infty) holds for all t>0. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.
Hilbert space von Neumann algebra semifinite trace \tau-measurable operator \tau-compact operator singular value function idempotent
Volga Region Mathematical Center
075-02-2022-882
Primary Language | English |
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Subjects | Mathematical Sciences |
Journal Section | Articles |
Authors | |
Project Number | 075-02-2022-882 |
Publication Date | March 15, 2023 |
Published in Issue | Year 2023 Volume: 6 Issue: 1 |