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Korovkin approximation of homomorphisms in involution algebras. (English) Zbl 0592.46051

The following Korovkin-type theorem is proved:
Let A be a Banach algebra with continuous involution and let S be a subset of A. Suppose that m is a self-adjoint Schwarz mapping on A such that m is multiplicative on the closure of the involutive subalgebra B generated by S. Then for every equicontinuous net \(\phi_ i\) of self- adjoint Schwarz mappings the property \[ \lim \phi_ i(x)=m(x)\quad \forall x\in S\cup SS^*\cup S^*S \] already implies \[ \lim \phi_ i(y)=m(y)\quad \forall y\in \bar B. \] Here \(SS^*\) denotes \(\{\) \(y\in A:\) \(y=xx^*\), \(x\in S\}\) and \(S^*S\) denotes \(\{\) \(y\in A:\) \(y=x^*x\), \(x\in S\}.\)
The theorem extends a result already given by B. V. Limaye and M. N. N. Namboodiri [see Theorem 1.2 in J. Indian Math. Soc., New Ser. 43(1980), 195-202 (1979; Zbl 0532.41025)]. However in contrast to that paper a new direct proof of the Korovkin-type theorem is developed.

MSC:

46K05 General theory of topological algebras with involution
41A36 Approximation by positive operators

Citations:

Zbl 0532.41025
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References:

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