## The upper browder spectrum in commutatively ordered banach algebras

##### Date

2021

##### Authors

Mumba, Sonick

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##### Publisher

The University of Zambia

##### 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.

##### Description

Thesis

##### Keywords

Algebra , Banach algebra