ON THE DIAGONALS OF PROJECTIONS IN MATRIX ALGEBRAS OVER VON NEUMANN ALGEBRAS Soumyashant Nayak Richard V. Kadison The main focus of this dissertation is on exploring methods to characterize the diagonals of projections in matrix algebras over von Neumann algebras. This may be viewed as a non-commutative version of the generalized Pythagorean theorem and its converse (Carpenter’s Theorem) studie...

M(a, fi) S M'(ah ft)Jf^(a,, ft). This is M. Riesz's fundamental theorem. (See M. Riesz [5] and a different proof in Paley [4] ; see also a generalization of the theorem in L. C. Young [ i l ] . ) M. Riesz's argument proved the convexity only in the triangle O ^ a ^ l , 0 ^ / 3 ^ 1 , a+f3*zl. The extension to the whole quadrant is due to Thorin [9]. We shall not give the proof of the theorem her...

Helly?s, Radon?s, and Caratheodory?s theorems are the basic of convex analysis have an important place. These been studied by different authors for classes convexity. theorem B-1 - sets has proved before Adilov Ye?ilce. In this article, Helly?s Radon?s discussed examined these sets.

Various function theorists have successfully defined and investigated different kinds of analytic functions. The applications such functions played significant roles in geometry theory as a field complex analysis. In this work, therefore, certain subclass univalent is using generalized differential operator we discussed \(TS_{\sigma, \delta} ^{~ \wp} (\vartheta ,\hbar ,\ell )\) with negative co...