Differential Banach *-algebras of compact operators associated with symmetric operators
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.
scientific article; zbMATH DE number 1213986