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)


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.


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 


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  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

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