Strobl22

Applied Harmonic Analysis and Friends

June 19th - 25th 2022

Strobl, AUSTRIA

"Regularity results for classes of Hilbert C*-modules with respect to special bounded C*-functionals"

Frank, Michael

Considering the deeper reasons of the appearance of a remarkable counterexample by J. Kaad and M. Skeide [13] we consider situations in which two Hilbert C*-modules M in N with orthogonal complement {0} over a fixed C*-algebra A of coeffcients cannot be separated by a non-trivial bounded A-linear functional r0 : N to A vanishing on M. In other words, the uniqueness of extensions of the zero functional from M to N is focussed. We show this uniqueness of extension for any such pairs of Hilbert C*-modules over W*-algebras, over monotone complete C*-algebras and over compact C*-algebras. Moreover, uniqueness of extension takes place also for any one-sided maximal modular ideal of any C*-algebra. Such a non-zero separating bounded A-linear functional r0 exist for a given pair of Hilbert C*-modules over agiven C*- algebra A iff there exists a bounded A-linear non-adjointable operator T0 : N to N such that the kernel of T0 is not biorthogonally closed w.r.t. N and contains M. This is a new perspective on properties of bounded modular operators that might appear in Hilbert C*-module theory. By the way, we find a correct proof of [9, Lemma 2.4] for monotone complete C*-algebras and for compact C*-algebras.
https://ps-mathematik.univie.ac.at/e/talks/strobl22_Frank_2022-06_MichaelFrank_Strobl22.pdf

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