## Document Type

Presentation Abstract

## Presentation Date

4-15-2013

## Abstract

Let *X* be a completely regular Hausdorff space and *A* be a complex commutative unital Banach algebra with norm ∥∥_{∞}. We denote by *C* (*X*, *A*) the unital algebra of all *A*-valued continuous functions on *X* with pointwise operations and unit element *e* (*x*)≡*e*, where e is the unit element of *A*. We denote by (*C*_{b} (*X*, *A*), ∥∥_{∞}) the subalgebra of *C* (*X*, *A*) of all bounded continuous functions, provided with the sup-norm ∥∥_{∞} on *X* given by

[Download the attached PDF file to see the equation here and the complete abstract.]

for every ƒ ∈*C*_{b}(*X*, *A*), and by (*C _{p}*(

*X*,

*A*), ∥∥

_{∞}) the subalgebra of all functions ƒ ∈

*C*(

_{b}*X*,

*A*) such that ̅ƒ ̅(̅

*X̅*) is compact in

*A*. It is easy to see that both are Banach algebras.

We study the maximal ideal space *M* (*(C _{b}*(

*X*,

*A*), ∥∥

_{∞})), invertibility in (

*C*(

_{b}*X*,

*A*), ∥∥

_{∞}) and establish necessary and sufficient conditions in order the set

*X*×

*M*(

*A*) to be dense in

*M*((

*C*(

_{b}*X*,

*A*), ∥∥

_{∞})) where

*M*(

*A*) is the maximal ideal space of

*A*.

## Recommended Citation

Arizmendi, Hugo, "On Banach Algebras of Bounded Continuous Functions with Values in a Banach Algebra" (2013). *Colloquia of the Department of Mathematical Sciences*. 424.

https://scholarworks.umt.edu/mathcolloquia/424

## Additional Details

Presented jointly with the Analysis Seminar.

Monday, 15 April 2013

3:10 p.m. in Math 103

4:00 p.m. Refreshments in Math Lounge 109