Function Theory Of Several Variables Research Articles

Non-commutative measure theory: Henkin and analytic functionals on $$\mathrm {C}^*$$-algebras

Henkin functionals on non-commutative $$\mathrm {C}^*$$ -algebras have recently emerged as a pivotal link between operator theory and complex function theory in several variables. Our aim in this paper is characterize these functionals through a notion of absolute continuity, inspired by a seminal theorem of Cole and Range. To do this, we recast the problem as a question in non-commutative measure theory. We develop a Glicksberg–König–Seever decomposition of the dual space of a $$\mathrm {C}^*$$ -algebra into an absolutely continuous part and a singular part, relative to a fixed convex subset of states. Leveraging this tool, we show that Henkin functionals are absolutely continuous with respect to the so-called analytic functionals if and only if a certain compatibility condition is satisfied by the ambient weak- $$*$$ topology. In contrast with the classical setting, the issue of stability under absolute continuity is not automatic in this non-commutative framework, and we illustrate its key role in sharpening our description of Henkin functionals. Our machinery yields new insight when specialized to the multiplier algebras of the Drury–Arveson space and of the Dirichlet space, and to Popescu’s noncommutative disc algebra. As another application, we make a contribution to the theory of non-commutative peak and interpolation sets.

Hidden harmony - geometric fantasies: the rise of complex function theory

The book is a history of complex function theory from its origins to 1914, when the essential features of the modern theory were in place. It is the first history of mathematics devoted to complex function theory, and it draws on a wide range of published and unpublished sources. In addition to an extensive and detailed coverage of the three founders of the subject – Cauchy, Riemann, and Weierstrass – it looks at the contributions of authors from d’Alembert to Hilbert, and Laplace to Weyl. Particular chapters examine the rise and importance of elliptic function theory, differential equations in the complex domain, geometric function theory, the early years of complex function theory in several variables. Unique emphasis has been devoted to the creation of a textbook tradition in complex analysis by considering some seventy textbooks in nine different languages. The book is not a mere sequence of disembodied results and theories, but offers a comprehensive picture of the broad cultural and social context in which the main actors lived and worked by paying attention to the rise of mathematical schools and of contrasting national traditions.

The dirichlet problem for a complex Monge-Amp�re equation

where G' denotes the Jacobian determinant of G. Furthermore, if G4=(0 . . . . . 0), then (d d ~ log h G I) 0. Thus, for f = 0, (1) is a natural generalization of the Dirichlet problem for harmonic in the complex plane. Other extended Dirichlet problems were studied in connection with function theory in several variables by S. Bergman [2, 3] (on domains with distinguished boundary surfaces) and more generally by H. Bremermann [4]. In Section 8 it is shown that the solution of the problem discussed by Bremermann actually solves (1), in a generalized sense, with f = 0 . The problem (1) seems to be a reasonable candidate for a (nonlinear) potential theory associated with the theory of of several complex variables. The question of uniqueness for the problem (1) is related to the question of existence of inner functions on the domain O. If h is a bounded analytic function

The Dirichlet problem for a complex Monge-Ampere equation

where G' denotes the Jacobian determinant of G. Furthermore, if G4=(0 . . . . . 0), then (d d ~ log h G I) 0. Thus, for f = 0, (1) is a natural generalization of the Dirichlet problem for harmonic in the complex plane. Other extended Dirichlet problems were studied in connection with function theory in several variables by S. Bergman [2, 3] (on domains with distinguished boundary surfaces) and more generally by H. Bremermann [4]. In Section 8 it is shown that the solution of the problem discussed by Bremermann actually solves (1), in a generalized sense, with f = 0 . The problem (1) seems to be a reasonable candidate for a (nonlinear) potential theory associated with the theory of of several complex variables. The question of uniqueness for the problem (1) is related to the question of existence of inner functions on the domain O. If h is a bounded analytic function