Differential Banach *-Algebras of Compact Operators Associated with Symmetric Operators

Abstract

Extensive development of noncommutative geometry requires elaboration of the theory of differential Banach *-algebras, that is, dense *-subalgebras of C*-algebras whose properties are analogous to the properties of algebras of differentiable functions. We consider a specific class of such algebras, D-algebras, and show that various *-algebras of compact operators associated with symmetric operators S on Hilbert spaces H are D-subalgebras of the C*-algebra of all compact operators C(H). We focus on how the properties of the operators S are reflected in the structure of these operator algebras.1998 Academic Press

On action of diffeomorphisms of C*-algebras on derivations

“…The *-derivations δ min S = δ S |F S and δ max S = δ S |J S of C(H ) are closed; they are the minimal and the maximal closed *-derivations of C(H ) with minimal implementation S. It was proved in [6] that the closure of (J S ) 2 in · δ S coincides with F S and that J S = F S if S is selfadjoint. In Section 3 we establish a link between minimal symmetric implementations of two derivations from Der(A).…”

“…Hence the norms · δ S and · δ coincide on A, so it follows from Lemma 3.1(ii) that F(A, δ) and F S are isometrically isomorphic. As F S is an ideal of A S (see [6]), F(A, δ) is an ideal of A.…”

“…It was shown in [6] that the algebra (F S , · δ S ) has a bounded approximate identity if and only if S is selfadjoint. By Corollary 3.2, F S = F T and the norms · δ S and · δ T on them are equivalent.…”

Second-Order Noncommutative Differential and Lipschitz Structures Defined by a Closed Symmetric Operator

“…Let K 1 S := A 1 S ∩ K(H), J 1 S := {A ∈ K 1 S : A S ∈ K(H)} and F 1 S be the closure in the norm • 1 of all finite rank operators in A 1 S . The algebra A 1 S is a Banach (D * 1 )-algebra [KS2] in the sense that it is a Banach * -algebra that is a dense * -subalgebra of a C * -algebra satisfying T R 1 ≤ T 1 R + T R 1 for all T, R in A 1 S . The algebras K 1 S , J 1 S , F 1…”

“…c 2015 Australian Mathematical Publishing Association Inc. 1446-7887/2015 $16.00 are closed subalgebras of (A 1 S , • 1 ) and F 1 S ⊂ J 1 S ⊂ K 1 S ⊂ A 1 S . In [KS2,KS3,KS4], Kissin and Shulman have investigated the structure of these algebras, regarding them as noncommutative differential algebras defined by the derivation δ S .…”