Семинар за анализу и примене, 14. децембар 2023.

Наредни састанак Семинара биће одржан онлајн у четвртак 14. децембра 2023. са почетком у 16 часова

Предавач: dr Michael Frank, Leipzig University of Technology, Economics and Culture Faculty of Computer Science and Media
Considering the deeper reasons of the appearance of a remarkable counterexample by J. Kaad and M. Skeide [17] we consider situations in which two Hilbert C*-modules M ⊂ N with M⊥ = {0} over a fixed C*-algebra A of coefficients cannot be separated by a non-trivial bounded A-linear functional r0 : N → A vanishing on M.  In other words, the uniqueness of extensions of the zero functional from M to N is focused.  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  M ⊆ N over a given C*-algebra A iff there exists a bounded A-linear non-adjointable operator  T0 : N → 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 [13, Lemma 2.4] in the case of monotone  complete and compact C*-algebras, but not in the general C*-case. Some ideas published by V. M. Manuilov in [31] need a thorough revision.  In the particular case, when N is the multiplier module of M both these modules might not coincide, but the orthogonal complement of M in N = V (M) equals to zero. One has an isometric inclusion of N0 into M0, sometimes even N0 ⊂ M0. So, non-trivial extensions of the zero modular functional on M to a non trivial bounded modular functional on N = V (M) vanishing on M cannot exist.

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