Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces
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arXiv
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| Format: | Preprint |
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2026
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| author | Bellomonte, Giorgia Ivkovic, Stefan Trapani, Camillo |
| author_facet | Bellomonte, Giorgia Ivkovic, Stefan Trapani, Camillo |
| contents | In this paper, a generalized Cauchy-Schwarz inequality for positive sesquilinear maps with values in noncommutative Lp-spaces for p > 1 are obtained. Bound estimates for their real and imaginary parts are also provided, and, as an application, a generalization of the uncertainty relation in the context of noncommutative L2-spaces are given. Next, a Cauchy-Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von Neumann algebra into a C*-algebra equipped with the numerical radius norm is proved. In the same spirit, a new norm on a noncommutative L2-space, which generalizes the classical numerical radius norm of bounded linear operators on a Hilbert space, is proposed, and a Cauchy-Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von-Neumann algebra into the noncommutative L2-space equipped with this new norm is proved. These results are used to get representations of general positive linear maps with values into a non-commutative Lp-space and into certain operator spaces in several different situations. Some concrete examples are also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11371 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces Bellomonte, Giorgia Ivkovic, Stefan Trapani, Camillo Operator Algebras Functional Analysis In this paper, a generalized Cauchy-Schwarz inequality for positive sesquilinear maps with values in noncommutative Lp-spaces for p > 1 are obtained. Bound estimates for their real and imaginary parts are also provided, and, as an application, a generalization of the uncertainty relation in the context of noncommutative L2-spaces are given. Next, a Cauchy-Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von Neumann algebra into a C*-algebra equipped with the numerical radius norm is proved. In the same spirit, a new norm on a noncommutative L2-space, which generalizes the classical numerical radius norm of bounded linear operators on a Hilbert space, is proposed, and a Cauchy-Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von-Neumann algebra into the noncommutative L2-space equipped with this new norm is proved. These results are used to get representations of general positive linear maps with values into a non-commutative Lp-space and into certain operator spaces in several different situations. Some concrete examples are also given. |
| title | Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2602.11371 |