Classify contractively complemented subspaces of general noncommutative Lp-spaces

Classify the contractively complemented subspaces of general noncommutative L^p-spaces, including the Jordan and ternary structures that may arise as ranges of contractive projections.

Background

The paper situates contractive complementation as a central structural problem in noncommutative L^p-theory. Although contractively complemented subspaces of classical L^p-spaces and Schatten spaces admit substantial classifications, the general noncommutative case is more complicated because ranges need not possess an associative algebra structure. The examples discussed in the introduction indicate that Jordan and ternary structures are natural and potentially unavoidable.

The results in the paper resolve important special cases, including 2-contractive projections on noncommutative L^4-spaces and finite-rank 2-contractive projections for general exponents. They do not provide a classification of all contractively complemented subspaces of arbitrary noncommutative L^p-spaces, leaving the broader structural problem unresolved.

References

The same work explicitly raises the problem of describing the contractively complemented subspaces of general noncommutative $L^p$-spaces. This problem remains one of the basic structural questions concerning noncommutative $L^p$-spaces.

— A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces (2610.03375 - Arhancet, 2 Oct 2026) in Section 1, Introduction

Classify contractively complemented subspaces of general noncommutative Lp-spaces

Background

References

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