The upper browder spectrum in commutatively ordered banach algebras

Abstract

A commutatively ordered Banach algebra (COBA) is a complex unital Banach algebra A containing a subset C, called an algebra c-cone, such that C contains the unit of A and is closed under addition, positive scalar multiplication and multiplication by commuting positive elements. If the commutativity assumption is removed, then the resulting cone is called an algebra cone. A Banach algebra ordered by an algebra cone is called an ordered Banach algebra (OBA): Evidently, every OBA is a COBA: Not every COBA is however an OBA: An example con rming this statement is given by the COBA (B(H);C); where B(H) is the space of all bounded linear operators on a Hilbert space H and C = fT 2 B(H) : hTx; xi 2 R+ for all x 2 Hg: Benjamin and Mouton described Fredholm theory in OBAs relative to a homo- morphism T : A ! B; where A and B are Banach algebras, and introduced an element called an upper Browder element. An upper Browder element x 2 A is an element of the form y + z; where y is invertible in A and z 2 C is an element of the null space of T such that yz = zy: We denote by B+ T the set of all upper Browder elements of A; which in turn gives (in a natural way) rise to the upper Browder spectrum + T (x) := f 2 C : e �� x is not an upper Browder elementg: In an OBA setting, Benjamin examined the following natural question: given that the spectral radius of a positive element is not in the Fredholm spectrum of the element, when will it be outside the upper Browder spectrum of that element? The element satisfying this condition is said to have the upper Browder spectrum property (see De nition 5.1.1). They went on to show that the connected hulls of the upper Browder and the Browder spectra do not coincide in general, as well as the conditions under which the upper Browder spectrum satisfy the spectral mapping theorem. In this study we extend these results to COBAs. Since every OBA is a COBA; it is actually concluded that some results on upper Browder spectrum of an OBA element readily extend to COBAs: For further research, we recommend that the COBAs; rather than the OBAs; should be the default setting for studying Fredholm theory.