## Annals of Functional Analysis

### Integral functionals on $C^*$-algebra of vector-valued regulated functions

#### Abstract

‎In this paper we deal with the notion of regulated functions with‎ ‎values in a $C^*$-algebra $\mathcal A$ and present examples using‎ ‎a special bi-dimensional $C^*$-algebra of triangular matrices‎. ‎We‎ ‎consider the Dushnik integral for these functions and shows that a‎ ‎convenient choice of the integrator function produces an integral‎ ‎homomorphism on the $C^*$-algebra of all regulated functions $‎ ‎G([a,b],\mathcal A)$‎. ‎Finally we construct a family of linear‎ ‎integral functionals on this $C^*$-algebra‎.

#### Article information

Source
Ann. Funct. Anal., Volume 3, Number 2 (2012), 21-31.

Dates
First available in Project Euclid: 12 May 2014

https://projecteuclid.org/euclid.afa/1399899930

Digital Object Identifier
doi:10.15352/afa/1399899930

Mathematical Reviews number (MathSciNet)
MR2948386

Zentralblatt MATH identifier
1261.46037

Subjects
Primary: 46K05: General theory of topological algebras with involution
Secondary: 45N05‎ ‎46C05

#### Citation

‎Fernandes, L‎. ‎A‎. ‎O‎.; ‎Arbach, R‎. Integral functionals on $C^*$-algebra of vector-valued regulated functions. Ann. Funct. Anal. 3 (2012), no. 2, 21--31. doi:10.15352/afa/1399899930. https://projecteuclid.org/euclid.afa/1399899930

#### References

• J. Dieudonné, Foundations of Modern Analysis, Academic Press, 1969.
• R.G. Douglas, Banach Algebra Techniques in Operator Theory, Springer, 1998.
• W. Arveson, An invitation to $C^*$-Algebras, Springer, 1976.
• L.A.O. Fernandes and R. Arbach, Regulated Functions with values in the Banach Algebra of Quaternions, ICAEM, World Congress on Engineering (WCE) 2011, London UK, vol 1, 196–201, 2011.
• L.A.O. Fernandes and R. Arbach, Integral Functionals on the Space of Regulated Functions with values in Banach Algebras, 8TH International Conference on Functions Spaces, Differential Operators and Nonlinear Analysis, FSDONA 2011, Tabarz/Thuringia, Germany, 2011.
• C.S. Hönig, Volterra-Stieltjes Integral Equations, Math. Studies 16, North Holland Publ. Company, 1975.
• W. Rudin, Functional Analysis: second edition, McGraw-Hill, 1991.