Differential properties of some dense subalgebras of C*-algebras

Abstract

The paper studies some classes of dense *-subalgebras B of C*-algebras A whose properties are close to the properties of the algebras of differentiable functions. In terms of a set of norms {||||,}f =1 on B it defines (DJ)-subalgebras of A and establishes that they are locally normal Q*-subalgebras. If x = x*eB and /(() is a function on Sp(x), some sufficient conditions are given for f(x) to belong to B. For p=\, in particular, it is shown that (D*)-subalgebras are closed under C"-calculus. If S is a closed derivation of A, the algebras D(5 P ) are (DJ)-subalgebras of A. In the case when 6 is a generator of a one-parameter semigroup of automorphisms of A, it is proved that, in fact, D(5 P ) are (Df)-subalgebras. The paper also characterizes those Banach '-algebras which are isomorphic to (D*)-subalgebras of C*-algebras.1991 Mathematics subject classification: 46H20, 46L57.

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

“…, it follows from Theorem 12(ii) [8] that ; n (x) # A and # n (x) # A. Applying Lemma 2.7, we obtain that &t 3 &# n (t)& Ä 0 and that…”

“…It was shown in [8] that if U has the identity then A is also a Q-subalgebra of U, that is, If U does not have an identity, it can be embedded in a larger C*-algebra U =C1+U with identity and with the norm &* 1+x&=|*| +&x&, * # C, x # U. Then the algebra A =C1+A, with the norm &* 1+x& 1 =|*| +&x& 1 , * # C, x # A, is a D-subalgebra of U .…”

“…In [1] and [8] it was shown that if A is a D-subalgebra of a C*-algebra U with identity then h(x) # A, for any x=x* # A and any…”

“…Blackadar and Cuntz [1] and the authors [8] introduced and investigated various classes of such algebras. In particular, in [8] where D 0 and where r(x*x) is the spectral radius of x*x in A. It was shown that these algebras can be equivalently described as dense article no.…”

Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras

“…Similarly, noncommutative affine algebraic geometry mainly deals with finitely generated algebras (without topology) and noncommutative projective algebraic geometry studies graded algebras. Finally, in noncommutative differential geometry one uses dense subalgebras of C * -algebras with some special "differential" properties (see [4,33]). …”