Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C-algebras. The stronger one of the two is called property (T) and the weaker one is called property (T e). It is shown that all non-unital C-algebras do not have property (T)(neither do their unitalizations). Moreover, all non-unital -unital C-algebras do not have property (T e).

Let E be a real Banach space with a uniformly Gteaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E,{T i} i= 1 N a finite family of asymptotically nonexpansive self-mappings on K with common sequence {k n} n= 1[1,),{t n},{s n} be two sequences in (0, 1) such that s n+ t n= 1 (n 1) and f be a contraction on K. Under suitable conditions on the sequences {s n},{t n}, we show the existence of a sequence {x n} satisfying the relation x n=(1 1 k n) x n+ s n k n f (x n)+ t n k n T r n n x n where n= l n N+ r n for some unique integers l n 0 and 1 r n N. Further we prove that {x n} converges strongly to a common fixed point of {T i} i= 1 N, which solves some variational inequality, provided x n T i x n 0 as n for i= 1, 2,, N. As an application, we prove that the iterative process defined by z 0 K, z n+ 1=(1 1 k n) z n+ s n k n f (z n)+ t n k n

Let A be an operator ideal on LCSs. A continuous seminorm p of a LCS X is said to be Acontinuous if Qp Ainj (X, Xp), where Xp is the completion of the normed space Xp= X/p 1 (0) and Qp is the canonical map. p is said to be a Groth (A)seminorm if there is a continuous seminorm q of X such that p q and the canonical map Qpq: Xq Xp belongs to A (Xq, Xp). It is well-known that when A is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infraSchwartz) space if and only if every continuous seminorm p of X is Acontinuous if and only if every continuous seminorm p of X is a Groth (A)seminorm. In this paper, we extend this equivalence to arbitrary operator ideals A and discuss several aspects of these constructions which are initiated by A. Grothendieck and D. Randkte, respectively. A bornological version of the theory is obtained, too.

Degree theory has been developed as a tool for checking the solution existence of nonlinear equations. In his classic paper published in 1983, Browder developed a degree theory for mappings of monotone type f+ T, where f is a mapping of class (S)+ from a bounded open set in a reflexive Banach space X into its dual X, and T is a maximal monotone mapping from X into X. This breakthrough paved the way for many applications of degree theoretic techniques to several large classes of nonlinear partial differential equations. In this paper we continue to develop the results of Browder on the degree theory for mappings of monotone type f+ T. By enlarging the class of maximal monotone mappings and pseudo-monotone homotopies we obtain some new results of the degree theory for such mappings.

Let <i>X</i> be a locally compact Hausdorff space and <i>C</i> ₀(<i>X</i>) the Banach space of continuous functions on <i>X</i> vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on <i>C</i> ₀(<i>X</i>). They arise from prime ideals of <i>C</i> ₀(<i>X</i>), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of <i>c</i> ₀ can be constructed explicitly through an ultrafilter on complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on <i>C</i> ₀(<i>X</i>) for arbitrary <i>X</i>. We also make some remarks where <i>C</i> ₀(<i>X</i>) is replaced by a non-commutative C*-algebra.