Topics in Operator Theory pp 509-518 | Cite as

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

Conference paper

## 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.
There exists a matrix

*g*∈*M*_{+}^{ k }×*n*such that*g***f*=*I*_{ k }. - 2.
There exist matrices

*F,G*∈*M*_{+}^{ 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

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