## Illinois Journal of Mathematics

### On the spectrum of Banach algebra-valued entire functions

#### Abstract

In this paper, we investigate a notion of spectrum $\sigma(f)$ for Banach algebra-valued holomorphic functions on $\mathbb{C}^{n}$. We prove that the resolvent $\sigma^{c}(f)$ is a disjoint union of domains of holomorphy when $\mathcal{B}$ is a $C^{\ast}$-algebra or is reflexive as a Banach space. Further, we study the topology of the resolvent via consideration of the $\mathcal{B}$-valued Maurer–Cartan type $1$-form $f(z)^{-1}\,df(z)$. As an example, we explicitly compute the spectrum of a linear function associated with the tuple of standard unitary generators in a free group factor von Neumann algebra.

#### Article information

Source
Illinois J. Math., Volume 55, Number 4 (2011), 1455-1465.

Dates
First available in Project Euclid: 12 July 2013

https://projecteuclid.org/euclid.ijm/1373636693

Digital Object Identifier
doi:10.1215/ijm/1373636693

Mathematical Reviews number (MathSciNet)
MR3082878

Zentralblatt MATH identifier
1273.47008

Subjects
Primary: 32A65: Banach algebra techniques [See mainly 46Jxx]
Secondary: 47A10: Spectrum, resolvent

#### Citation

Bannon, J. P.; Cade, P.; Yang, R. On the spectrum of Banach algebra-valued entire functions. Illinois J. Math. 55 (2011), no. 4, 1455--1465. doi:10.1215/ijm/1373636693. https://projecteuclid.org/euclid.ijm/1373636693

#### References

• H. Bart, T. Ehrhardt and B. Silbermann, Trace conditions for regular spectral behavior of vector-valued analytic functions, Linear Algebra Appl. 430 (2009), 1945–1965.
• S. S. Chern, Complex manifolds without potential theory, 2nd ed., Springer-Verlag, New York, 1979.
• K. Davidson, $C^{\ast}$-algebra by examples, American Mathematical Society, Providence, RI, 1996.
• R. G. Douglas, Banach algebra techniques in operator theory, 2nd ed., Springer, New York, 1998.
• I. C. Gohberg and E. I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Roché, Math. Sb. 13 (1971), 603–625.
• U. Haagerup and F. Larsen, Brown's spectral distribution measure for R-diagonal elements in finite von Neumann algebras, J. Funct. Anal. 176 (2000), 331–367.
• R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, vols. I & II, Academic Press, London, 1983 & 1986.
• D. Voiculescu, K. J. Dykema and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, 1992.
• R. Yang, Projective spectrum in Banach algebras, J. Topol. Anal. 1 (2009), 289–306.