Abstract
We study ideal F-norms ‖·‖p, 0 <p < +∞ associated with a trace ϕ on a C*-algebra \({\cal A}\). If A, B of \({\cal A}\) are such that |A|≤ |B|,then ‖A‖p ≤ ‖B‖p. We have ‖A‖p = ‖A*‖p for all A from \({\cal A}\) (0 <p < +∞)and a seminorm ‖·‖p for 1 ≤ p< +∞. Weestimate the distance from any element of a unital \({\cal A}\) to the scalar subalgebra in the seminorm ‖·‖1. We investigate geometric properties of semiorthogonal projections from \({\cal A}\). If a trace φ is finite, then the set of all finite sums of pairwise products of projections and semiorthogonal projections (in any order) of \({\cal A}\) with coefficients from ℝ+ is not dense in \({\cal A}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bikchentaev, A. M. “Ideal F-Norms on C*-Algebras”, Russian Mathematics 59, No. 5, 58–63 (2015).
Bikchentaev, A. M. “Inequality for a Trace on a Unital C*-Algebra”, Math. Notes 99, No. 4, 487–491 (2016).
Bikchentaev, A. M. “On the Representation of Elements of a von Neumann Algebra in the Form of Finite Sums of Products of Projections. III. Commutators in C*-Algebras”, Sb. Math. 199, No. 3–4, 477–493 (2008).
Murphy, G. J. C*-Algebras and Operator Theory (Academic Press, 1990; Faktorial, Moscow, 1997).
Gross, J., Trenkler, G., Troschke, S.-O. “On Semi-Orthogonality and a Special Class of Matrices”, Linear Algebra Appl. 289, No. 1–3, 169–182 (1999).
Bikchentaev, A. M. “Semi-Orthogonal Projectors in a Hilbert Space”. in: At the Turn of the 20th-21st Centuries (Kazan Mat. Soc., Kazan, 2003), pp. 108–114 [in Russian].
Akemann, C. A., Anderson, J., Pedersen, G. K. “Triangle Inequalities in Operator Algebras”, Linear Multilinear Algebra 11, No. 2, 167–178 (1982).
Dixmier, J. Les C*-Algèbres et Leurs Représentations (Gauthier-Villars, Paris, 1969; Nauka, Moscow, 1974).
Blackadar, B. “A Simple Unital Projectionless C*-Algebra”, J. Operator Theory 5, No. 1, 63–71 (1981).
Kadison, R., Pedersen, G. “Means and Convex Combinations of Unitary Operators”, Math. Scand. 57, No. 2, 249–266 (1985).
Haagerup, U. “On Convex Combinations of Unitary Operators in C*-Algebras”. in: Progress in Math. Vol. 84. ‘Mappings of operator algebras’ (Birkhäuser, Boston, 1990), pp. 1–13.
Marcoux, L. W. “Projections, Commutators and Lie Ideals in C*-Algebras”, Math. Proc.R. Ir. Acad. 110A, No. 1, 31–55 (2010).
Bikchentaev, A. M. “On the Representation of Eements of a von Neumann Algebra in the Form of Finite Sums of Products of Projections”, Siberian Math. J. 46, No. 1, 24–34 (2005).
Acknowledgments
Supported by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (1.9773.2017/8.9).
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © A.M. Bikchentaev, 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 3, pp. 90–96.
About this article
Cite this article
Bikchentaev, A.M. Ideal F-Norms on C*-Algebras. II. Russ Math. 63, 78–82 (2019). https://doi.org/10.3103/S1066369X19030071
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X19030071