In the present paper, among other results, a decomposition formula is given for the w-bounded continuous negative definite functions of a topological *-semigroup S with a weight function w into a proper H*-algebra A in terms of w-bounded continuous positive definite A-valued functions on S. A generalization of a well-known result of K. Harzallah is obtained. An earlier conjecture of the author ...

This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or σ-complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

We develop a notion of derivative of a real-valued function on a Banach space, called the L-derivative, which is constructed by introducing a generalization of Lipschitz constant of a map. As with the Clarke gradient, the values of the L-derivative of a function are non-empty weak* compact and convex subsets of the dual of the Banach space. The L-derivative, however, is shown to be upper semi c...

This paper refers to the theory of semirings continuous numerical functions, which has been developed within functional algebra. The object investigation is CP(X) partial functions on topological spaces X with values in field R real numbers. subject study subalgebras CP(X). Some properties lattices A(X) all possible and A1(X) identity are considered. structure atoms preatoms clarified. allowed ...