Recent zbMATH articles in MSC 22Dhttps://zbmath.org/atom/cc/22D2021-06-15T18:09:00+00:00WerkzeugApproximate biflatness and approximate biprojectivity of some Banach algebras.https://zbmath.org/1460.460702021-06-15T18:09:00+00:00"Sahami, A."https://zbmath.org/authors/?q=ai:sahami.amirLet \(A\) be a Banach algebra, \(1_A:A\to A\) the identity map on \(A\), \(A\otimes_p A\) the projective tensor product of \(A\) with \(A\) and \(\pi_A:A\otimes_p A\rightarrow A\) the product morphism, defined by \(\pi_A(a\otimes_p b)=ab\) for every \(a, b\in A\). A Banach algebra \(A\) is called approximately biflat (respectively, approximately biprojective) if there exists a net of \(A\)-bimodule morphisms \(\rho_\alpha: (A\otimes_pA)^{*}\rightarrow A^*\) (respectively, \(\rho_\alpha:A\rightarrow A\otimes_pA\)) such that the net \((\rho_\alpha\circ \pi_A^{*}\)) converges to \(1_A\) in the weak star operator topology (respectively, the net \((\pi_A\circ\rho_\alpha)\) converges to \(1_A\) in the norm topology).
Let \(A\) be a Banach algebra and \(\phi\) a character of \(A\). If there exists an element \(a_0\in A\) such that \(aa_0=a_0a\) and \(\phi(a_0)=1\) for each \(a\in A\), then \(a\) is said to be a \(\phi\)-commutative element of \(A\). \(A\) is called approximately left \(\phi\)-amenable if there exists a net \((a_\alpha)\) such that the net \((aa_\alpha-\phi(a)a_\alpha)\) converges to the zero element of \(A\) and the net \((\phi(a_\alpha))\) converges to \(1\) for every \(a\in A\).
The authors show that
\begin{itemize}
\item[(a)] every approximately biflat Banach algebra is approximately biprojective;
\item[(b)] if \(A\) is a Banach algebra which has a \(\phi\)-commutative element for some character \(\phi\) of \(A\) and \(A^{**}\) is an approximately biprojective Banach algebra, then \(A\) is approximately left \(\phi\)-amenable.
\end{itemize}
The authors also find some sufficient conditions for a triangular Banach algebra that guarantee that this triangular algebra is not approximately biflat or approximately biprojective.
The results, obtained in the preceding sections, are then applied to some Banach algebras related to a locally compact group, more precisely, for classical Segal algebras \(S(G)\subseteq L^1(G)\), limiting (in some results) only to the cases when \(G\) is a SIN group or an Abelian group.
Reviewer: Mart Abel (Tartu)Lectures on representations of locally compact groups.https://zbmath.org/1460.220012021-06-15T18:09:00+00:00"Colojoară, Ion"https://zbmath.org/authors/?q=ai:colojoara.ion"Gheondea, Aurelian"https://zbmath.org/authors/?q=ai:gheondea.aurelianPublisher's description: This is a modern presentation of the theory of representations of locally compact groups. In a small number of pages, the reader can get some of the most important theorems on this subject. Many examples are provided. Highlights of the volume include:
(1) A generous introduction explaining the origins of group theory and their representations, the motivation for the main problems in this theory, and the deep connections with modern physics.
(2) A solid presentation of the theory of topological groups and of Lie groups.
(3) Two proofs of the existence of Haar measures.
(4) The detailed study of continuous representations on general locally convex spaces, with an emphasis on unitary representations of compact groups on Hilbert spaces.
(5) A careful presentation of induced representations on locally convex spaces and G. W. Mackey's Imprimitivity Theorem.
About half of the results included in this volume appear for the first time in a book, while the theory of \(p\)-induced representations on locally convex spaces is new. To facilitate reading, several appendices present the concepts and basic results from general topology, differential manifolds, abstract measures and integration, topological vector spaces, Banach spaces, Banach algebras, \(C^*\)-algebras, and operator theory on Hilbert spaces.\(C^*\)-algebras and their automorphism groups. Edited by Søren Eilers and Dorte Olesen. 2nd edition.https://zbmath.org/1460.460012021-06-15T18:09:00+00:00"Pedersen, Gert K."https://zbmath.org/authors/?q=ai:pedersen.gert-kjaergardPublisher's description: This elegantly edited landmark edition of Gert Kjærgård Pedersen's \(C^*\)-Algebras and their Automorphism Groups (1979; Zbl 0416.46043) carefully and sensitively extends the classic work to reflect the wealth of relevant novel results revealed over the past forty years. Revered from publication for its writing clarity and extremely elegant presentation of a vast space within operator algebras, Pedersen's monograph is notable for reviewing partially ordered vector spaces and group automorphisms in unusual detail, and by strict intention releasing the $C^*$-algebras from the yoke of representations as Hilbert space operators. Under the editorship of Søren Eilers and Dorte Olesen, the second edition modernizes Pedersen's work for a new generation of \(C^*\)-algebraists, with voluminous new commentary, all-new indexes, annotation and terminology annexes, and a surfeit of new discussion of applications and of the author's later work.