An Alternative Approach to Weighted Non-commutative Banach Function Spaces

Abstract

We use a weighted analogue of a trace to define a weighted noncommutative decreasing rearrangement and show it's relationship with the singular value function. We further show an alternative approach to constructing weighted non-commutative Banach function spaces using weighted non-commutative decreasing rearrangements and prove that this approach is equivalent to the original approach by Labuschagne and Majewski.

Weighted Noncommutative Banach Function Spaces

“…For the final result in this section, we will show that we can find yet another way of computing µ t (a, x). This result was proved in [15,Theorem 3.7].…”

“…The results in Lemma 3.4 and Corollary 3.5 below can be arrived at by an application of Theorem 2.25. We believe, however, that the arguments given in [15] are more revealing of the underlying nature of these results. As such we will first give the proofs as they appeared in [15], after which we will show how one can arrive at the these results by simply applying Theorem 2.25.…”

“…We believe, however, that the arguments given in [15] are more revealing of the underlying nature of these results. As such we will first give the proofs as they appeared in [15], after which we will show how one can arrive at the these results by simply applying Theorem 2.25. We now show how one can arrive at these results by an application of Theorem 2.25.…”

“…Section Notes. The majority of the results in this section first appeared in [15]. Exceptions are Lemmas 2.21, 2.22 and 2.23.…”

“…Proposition 2.3 was proved in [10] by Labuschagne and Majewski. Theorem 2.10 was proved in [15] by the author and Theorem 2.12 was proved in [12] by Labuschagne and the author.…”