Research in the noncommutative dynamical systems through the method of functional analysis together with the research in the interplay between topological dynamical systems and operator theory

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Operator algebras corresponding to topological dynamical systems have been studied as C-clossed products. On the other hand, there are Banach-algebra clossed products. These differences are the existence of non-self sdjoint closed ideal. One of the main results is the following : Every closed ideal of l^1 is self adjoint if and only if its dynamical system is free. Commutants of algebras are commutative maximal Banach-algebras and the intersection of closed ideals is non empty. We proved it for Banach-algebras and showed the existence projections to the maximal abelian subalgebras. As a more closing Are semi-hyponormal operators convexoid? This is a open problem for 20 years. About this problem, we could show that if, for the unilateral shift U, put S=T^2, then S is convexoid. It is published from the journal Linear and Multilinear Algebra with title"A remark on numerical range of semi-hyponormal operators". Next we studied Polaroid operators on a Banach space. We showed that if Polaroid operators have single value extension property, then Weyl's Theorem holds for these operators. Also we showed that if an operator T is quasi-similar with Bishop property and Polaroid, then T is Polaroid. These results are published from Journal of Mathematical Analysis and Applications.