Differential Banach *-algebras of compact operators associated with symmetric operators
type: Item
instance of: scholarly article
title: Differential Banach *-algebras of compact operators associated with symmetric operators
zbMATH Open document ID: 0936.46042
author: E. Kissin
author: V. S. Shulman
published in: Journal of Functional Analysis
publication date: 1998-12-02T00:00:00Z
review text: A special class of differential Banach \(\ast\)-algebras, called \(D\)-algebras, are extensively studied (for the case \(D=1\)) in the paper. Such an algebra can be identified with the norm dense Banach \(\ast\)-subalgebra \((A, \|\cdot\|_1)\) of \(C^\ast\)-algebra \((C,\|\cdot\|)\) satisfying the inequality \[ \|xy\|_1\leq \|x\|\|y\|_1+\|x\|_1\|y\|. \] Among others it is shown that \(\overline{A^n}=\overline{A^2}\) for all \(n\geq 2\) and that, generally speaking, \(\overline{A^2}\not= A\). Let \(S\) be a closed symmetric operator on a Hilbert space \(H\). Then \(S\) implements the derivation \(\delta_S\) on the algebra \[ A_S=\{ A\in B(H)\mid \;AD(S)\subset D(S), A^\ast D(S)\subset D(S); \] \[ (SA-AS)\mid D(S)\text{\;extends\;to\;a\;bounded\;operator}\}. \] It turns out that \((A_S, \|\cdot\|_1)\) becomes a D-algebra (\(D=1\)), where \(\|x\|_1=\|x\|+\|\delta(A)\|. \) The main theme of the paper is the relationship between the properties of the operator \(S\) and the structure of the following D-algebras of compact operators (here \(C(H)\) denotes the algebra of all compact operators on \(H\)); \( K_S=A_S\cap C(H)\), \(J_S=\{a\in K_S\mid \delta_S(a)\) is compact\}, and \(F_S=\|\cdot\|_1\) -- closure of finite rank operators in \(A_S.\) For example, it is shown that \(S\) is selfadjoint if and only if \(J_S\) and \(F_S\) have a bounded approximate unit. Further, a one-to-one correspondence between the essential left ideals of \(F_S\) and the closed subspaces of \(D(S)\) (equipped with the scalar product \(\langle x,y\rangle=(x,y)+(Sx,Sy)\)) is established. Many other results on ideals in D-algebras of compact operators are proved.
reviewed by: Jan Hamhalter
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L57
zbMATH DE Number: 1213986
zbMATH Keywords: differential Banach \(\ast\)-algebras
zbMATH Keywords: symmetric operators
zbMATH Keywords: derivation on \(C^\ast\)-algebras
zbMATH Keywords: \(D\)-algebras of compact operators
MaRDI profile type: Publication
full work available at URL: jfan.1997.3186
OpenAlex ID: W2084885006
cites work: Q5287655
cites work: Unbounded derivations of \(C^*\)-algebras
cites work: Q5345208
cites work: Q5723435
cites work: Q5526485
cites work: Indices of unbounded derivations of \(C^*\)-algebras
cites work: DenseQ-subalgebras of Banach andC*-algebras and unbounded derivations of Banach andC*-algebras
cites work: Differential properties of some dense subalgebras of C*-algebras
cites work: A remark on the domain of an unbounded derivation of a \(C^*\)-algebra
cites work: Q3997905
DOI: 10.1006/JFAN.1997.3186
description: scientific article; zbMATH DE number 1213986