Advertisement

Extension to an Invertible Matrix in Convolution Algebras of Measures Supported in [0,+∞)

  • Amol Sasane
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

Let M +denote the Banach algebra of all complex Borel measures with support contained in [0,+8), with the usual addition and scalar multiplication, and with convolution *, and the norm being the total variation of μ. We show that the maximal ideal space X(M +) of M +, equipped with the Gelfand topology, is contractible as a topological space. In particular, it follows that every left invertible matrix with entries from M +can be completed to an invertible matrix, that is, the following statements are equivalent for f ∈ (M +) n×k , k < n:
  1. 1.

    There exists a matrix gM + k ×n such that g * f = I k .

     
  2. 2.

    There exist matrices F,GM + n×n such that G * F = I n and F ij = f ij , 1 < i = n, 1 < j < k.

     

We also show a similar result for all subalgebras of M + satisfying a mild condition.

Keywords

Contractibility of the maximal ideal space convolution algebra of measures Hermite ring Tolokonnikov’s lemma coprime factorization 

Mathematics Subject Classification (2000)

Primary 54C40 Secondary 46J10 32A38 93D15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Hille and R.S. Phillips. Functional analysis and semi-groups. Third printing of the revised edition of 1957. American Mathematical Society Colloquium Publications, Vol. XXXI. American Mathematical Society, Providence, R.I., 1974.Google Scholar
  2. [2]
    V.Ya. Lin. Holomorphic fiberings and multivalued functions of elements of a Banach algebra. Functional Analysis and its Applications, no. 2, 7:122–128, 1973, English translation.Google Scholar
  3. [3]
    K.M. Mikkola and A.J. Sasane. Bass and Topological Stable Ranks of Complex and Real Algebras of Measures, Functions and Sequences. To appear in Complex Analysis and Operator Theory.Google Scholar
  4. [4]
    A.J. Sasane. The Hermite property of a causal Wiener algebra used in control theory. To appear in Complex Analysis and Operator Theory.Google Scholar
  5. [5]
    M. Vidyasagar. Control System Synthesis: A Factorization Approach. MIT Press Series in Signal Processing, Optimization, and Control, 7, MIT Press, Cambridge, MA, 1985.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Amol Sasane
    • 1
  1. 1.Mathematics DepartmentLondon School of EconomicsLondonUK

Personalised recommendations