On the spectrum of Banach algebra-valued entire functions. (English) Zbl 1273.47008
Summary: 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.
MSC:
| 47A10 | Spectrum, resolvent |
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 32A65 | Banach algebra techniques applied to functions of several complex variables |
| 46H99 | Topological algebras, normed rings and algebras, Banach algebras |
References:
| [1] | 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. · Zbl 1188.47014 · doi:10.1016/j.laa.2008.11.004 |
| [2] | S. S. Chern, Complex manifolds without potential theory , 2nd ed., Springer-Verlag, New York, 1979. · Zbl 0444.32004 |
| [3] | K. Davidson, \(C^{\ast}\)-algebra by examples , American Mathematical Society, Providence, RI, 1996. |
| [4] | R. G. Douglas, Banach algebra techniques in operator theory , 2nd ed., Springer, New York, 1998. · Zbl 0920.47001 · doi:10.1007/978-1-4612-1656-8 |
| [5] | 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. · Zbl 0254.47046 · doi:10.1070/SM1971v013n04ABEH003702 |
| [6] | 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. · Zbl 0984.46042 · doi:10.1006/jfan.2000.3610 |
| [7] | R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras , vols. I & II, Academic Press, London, 1983 & 1986. · Zbl 0888.46039 |
| [8] | D. Voiculescu, K. J. Dykema and A. Nica, Free random variables , CRM Monograph Series, vol. 1, American Mathematical Society, Providence, 1992. · Zbl 0795.46049 |
| [9] | R. Yang, Projective spectrum in Banach algebras , J. Topol. Anal. 1 (2009), 289-306. · Zbl 1197.47015 · doi:10.1142/S1793525309000126 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
