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Segal extensions and Segal algebras in uniform Banach algebras. (English) Zbl 1481.46040

A Segal extension of a Banach algebra \((A, \| \cdot\|_A)\) is a pair \(((B, \| \cdot\|_B), \phi)\) that satisfies the following conditions:
(1)
\((B, \| \cdot\|_B)\) is a Banach algebra;
(2)
\(\phi:A\rightarrow B\) is a continuous algebra homomorphism;
(3)
\(\phi(A)\) is a dense two-sided ideal of \(B\).
In more modern terminology, it means that the triple \((A, \phi, B)\) is a two-sided Segal Banach algebra (in case one considers arbitrary topological algebras \((A, \tau)\) and \((B, \rho)\) instead of \((A, \|\cdot\|_A)\) and \((B, \| \cdot\|_B)\), then one obtains the definition of a two-sided Segal topological algebra).
It is known that, for a fixed topological algebra (in particular, a Banach algebra) \(A\), there could exist different choices of a topological algebra \(B\) and a continuous algebra homomorphism \(\phi: A\rightarrow B\) such that \((A, \phi, B)\) is a two-sided Segal topological algebra (in particular, a two-sided Segal Banach algebra).
Given a Banach algebra \((A, \|\cdot\| )\), the paper tries to describe different Banach algebras \((B, \|\cdot\| )\) and continuous algebra homomorphisms \(\phi:A\rightarrow B\) such that \(((B, \|\cdot\| _B), \phi)\) is a Segal extension of \((A, \|\cdot\| _A)\).
For a Banach algebra \((A, \|\cdot\| _A)\),
(1)
the multiplier seminorm \(\|\cdot\| _M\) on \(A\) is defined by \[ \|a\| _M=\sup_{b\in A, \|b\| _A\leqslant 1}\{\|ab\| _A, \|ba\| _A\} \] for each \(a\in A\);
(2)
a Segal seminorm \(p\) of type I on \(A\) is an algebra seminorm on \(A\) such that there exist constants \(k, l>0\) such that \(k\|a\| _A\leqslant p(a)\leqslant l\|a\|_A\) for all \(a\in A\).
The first main result of the paper claims the following:
Let \((A, \|\cdot\| _A)\) be a Banach algebra and \(I=\{a\in A : \|a\| _M=0\}\). Then the following hold.
(i)
The Hausdorff completion \(({(A/I, \|\cdot\| _{M}^{\sim})}^{\sim}, \kappa_I)\), of \(A\), where \(\kappa_I:A\rightarrow A/I\) is the quotient map, is a Segal extension of \((A, \|\cdot\| _A)\).
(ii)
If \(((B, \|\cdot\| _B), \phi)\) is a Segal extension of \((A, \|\cdot\|_A)\), where the norm \(\|\cdot\| _B\) is induced by a Segal seminorm of type I, then there exists an algebra homomorphism from \((B, \|\cdot\| _B)\) to \((A/I, \|\cdot\| _{M}^{\sim})^{\sim})\) with dense range.
The authors also find necessary and sufficient conditions under which the pair \(((C_0(\Delta(A)), \tau_{C_0(\Delta(A))}, \Phi)\), where \(\Delta(A)\) is the Gelfand space of \(A\) and \(\Phi:A\rightarrow C_0(\Delta(A))\) is the Gelfand map, is a Segal extension of a commutative Banach algebra \((A, \|\cdot\| _A)\).
A Banach algebra \((A, \|\cdot\| _A)\) is a uB-Segal algebra if it has a Segal extension \(((B, \|\cdot\| _B), i)\), where \((B, \|\cdot\| _B)\) is a uniform Banach algebra (i.e., \(\|b^2\| _B= \|b\|^2 _B\) for each \(b\in B\)) and \(i:A\rightarrow B\) is also injective.
The last section of the paper deals with properties of uB-Segal algebras. For example, the authors prove that a faithful (i.e., \(x\in A\), \( xA=\{\theta_A\}\) implies that \(x=\theta_A\)) commutative Banach algebra \((A, \|\cdot\| _A)\) is a uB-Segal algebra if and only if \(A\) is semisimple and there exists \(l>0\) such that \(\max\{\|ab\| _A, \|ba\| _A\}\leqslant l \,r_A(a)\|b\| _A\) for all \(a, b\in A\), where \(r_A(a)\) denotes the spectral radius of \(a\).
Reviewer’s remarks: The paper contains several editorial errors (subscripts in places they should not be, probably errors in placing curly brackets correctly in LATEX). Also, the authors use different terminology for the same notion throughout the paper.
Reviewer: Mart Abel (Tartu)

MSC:

46H05 General theory of topological algebras
46K05 General theory of topological algebras with involution
46J05 General theory of commutative topological algebras
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References:

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