Recent zbMATH articles in MSC 46Hhttps://www.zbmath.org/atom/cc/46H2021-04-16T16:22:00+00:00WerkzeugExtending representations of Banach algebras to their biduals.https://www.zbmath.org/1456.460422021-04-16T16:22:00+00:00"Gardella, Eusebio"https://www.zbmath.org/authors/?q=ai:gardella.eusebio"Thiel, Hannes"https://www.zbmath.org/authors/?q=ai:thiel.hannesLet \(\varphi\) be a representation of a Banach algebra \(A\) on a Banach space \(X\). The essential space \(X_\varphi\) of \(\varphi\) is the closed linear span of \{\(\varphi(a)x$,\, $a \in A$,\, $x \in X\)\}. Firstly, the authors give conditions so that there is an extension \(\widetilde{\varphi}\) of \(\varphi\) on the bidual (Banach) algebra of \(A\), under (the left or yet the right) Arens product; in that case, the essential spaces of \(\varphi\) and \(\widetilde{\varphi}\) agree. Based on this result, the authors deal with the complementarity of the essential spaces (with respect to representations of Banach algebras). A positive result is given when \(A\) has a bounded left approximate identity and every operator \(A \rightarrow X\) is weakly compact.
Further, the authors connect the existence of representations of certain Banach algebras to that of the respective (left) multiplier algebras. In particular, they consider the case when \(X\) belongs to a class of reflexive Banach spaces, closed under complementation (of its subspaces) and \(A\) is a certain representable Banach algebra with a contractive left approximate identity. Actually, \(A\) accepts a nondegenerate isometric representation on some space like \(X\). This implies that the left multiplier algebra of \(A\) has a unital,
isometric representation on a space \(X\), as before. This is also true for any \(C^\ast\)-algebra being isometrically representable on an \(L^p\)-space, \(p \in [1, \infty)\). Based, amongst others, on the latter result, the authors prove that, in the context of \(C^\ast\)-algebras \(A\), the (ring) commutativity characterizes such an \(A\) as isometrically represented on an \(L^p\)-space, \(p \in [1, \infty) \backslash \{2\}\).
Moreover, for a locally compact group \(G\), the completion of \(L^1(G)\) for nondegenerate representations
on \(L^p\)-spaces is called the universal group \(L^p\)-operator algebra of \(G\). Actually, the latter is universal for all contractive representations of \(L^1(G)\) on \(L^p\)-spaces.
Applications of the present paper are given by the authors in [Trans. Am. Math. Soc. 371, 2207--2236 (2019; Zbl 06999077)].
Reviewer: Marina Haralampidou (AthÃna)Hyperreflexivity constants of the bounded \(n\)-cocycle spaces of group algebras and \(C^*\)-algebras.https://www.zbmath.org/1456.460452021-04-16T16:22:00+00:00"Samei, Ebrahim"https://www.zbmath.org/authors/?q=ai:samei.ebrahim"Farsani, Jafar Soltani"https://www.zbmath.org/authors/?q=ai:farsani.jafar-soltaniSummary: We introduce the concept of strong property \((\mathbb{B})\) with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all \(C^*\)-algebras and group algebras have the strong property \((\mathbb{B})\) with a constant given by \(288 \pi (1 + \sqrt{2})\). We then use this result to find a concrete upper bound for the hyperreflexivity constant of \(\mathcal{Z}^n (A, X)\), the space of bounded \(n\)-cocycles from \(A\) into \(X\), where \(A\) is a \(C^*\)-algebra or the group algebra of a group with an open subgroup of polynomial growth and \(X\) is a Banach \(A\)-bimodule for which \(\mathcal{H}^{n+1} (A, X)\) is a Banach space. As another application, we show that for a locally compact amenable group \(G\) and \(1 < p < \infty\), the space \(CV_P(G)\) of convolution operators on \(L^p(G)\) is hyperreflexive with a constant given by \(384 \pi^2 (1 + \sqrt{2})\). This is the generalization of a well-known result of \textit{E. Christensen} [Math. Scand. 50, 111--122 (1982; Zbl 0503.47032)] for \(p = 2\).Pseudospectrum of an element of a Banach algebra.https://www.zbmath.org/1456.470022021-04-16T16:22:00+00:00"Krishnan, Arundhathi"https://www.zbmath.org/authors/?q=ai:krishnan.arundhathi"Kulkarni, S. H."https://www.zbmath.org/authors/?q=ai:kulkarni.s-hSummary: The \(\varepsilon\)-pseudospectrum \(\Lambda_\varepsilon(a)\) of an element \(a\) of an arbitrary Banach algebra \(A\) is studied. Its relationships with the spectrum and numerical range of \(a\) are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of \(a\). Suppose for some \(\epsilon > 0\) and \(a,b\in A\), \(\Lambda_\epsilon(ax) = \Lambda_\epsilon(bx)\) for all \(x \in A\). It is shown that \(a=b\) if:
\begin{itemize}
\item[(i)] \(a\) is invertible.
\item[(ii)] \(a\) is Hermitian idempotent.
\item[(iii)] \(a\) is the product of a Hermitian idempotent and an invertible element.
\item[(iv)] \(A\) is semisimple and \(a\) is the product of an idempotent and an invertible element.
\item[(v)] \(A=B(X)\) for a Banach space \(X\).
\item[(vi)] \(A\) is a \(C^*\)-algebra.
\item[(vii)]\(A\) is a commutative semisimple Banach algebra.
\end{itemize}