On commutativity of C*-algebras

Abstract

1. Two numerical characterizations of commutativity for C*-algebra si (acting on the Hilbert space H) were given in [1]; one used the norms of self-adjoint operators in si (Theorem 2), and the other the numerical index of si (Theorem 3). In both cases the proofs were based on the result of Kaplansky which states that if the only nilpotent operator in si is 0, then si is commutative ([2] 2.12.21, p. 68). Of course the converse also holds.We shall apply in this note both Kaplansky's result and Holbrook's operator radii [3] to give two types of characterizations; one is by means of operator radii, and the other in terms of C p -classes of operators in si. These also enable us to generalize Theorem 2 and 3 in [1]. Finally, a particular case of our Theorem 5 shows that si is commutative if and only if every T esi satisfies the first order growth condition (Gi).2. First we need some notation, definitions and well known results. Let T e si. We recall that T is in the class C p (p > 0), operators having unitary p-dilation, if

Commutativity ofC*-algebras and associativity ofJB*-algebras

“…Let A be a unital C*-algebra acting on a Hilbert space H. As in [27], (1-111), for any p > 0, define the class C p (A) as the set of all aeA such that there exists some Hilbert space K containing H as a subspace, satisfying a n = pP H (u n ) for all natural numbers n and a suitable unitary operator u on K (P H denotes the orthogonal projection from K onto H). Then Also all properties of w p listed in [24] for C*-algebras remain true for n.c. JB*algebras. In particular, if 6 2 = 0, then w p (b) = p" 1 !!^!!…”

“…Moreover some of the above results can be sharpened using generalized notions of numerical radius and index, and the counterpart for JB*-algebras of the commutativity criteria for C*-algebras in [24] can be obtained, as follows. Let A be a unital C*-algebra acting on a Hilbert space H. As in [27], (1-111), for any p > 0, define the class C p (A) as the set of all aeA such that there exists some Hilbert space K containing H as a subspace, satisfying a n = pP H (u n ) for all natural numbers n and a suitable unitary operator u on K (P H denotes the orthogonal projection from K onto H).…”

“…Several papers have been devoted to the characterization of commutativity in C*-algebras ( [12,16,24]; see [14], p. 287 and [4], chapter 2). Most of them rely on Kaplansky's theorem stating that a C*-algebra is commutative if and only if it has no non-trivial nilpotent elements (see [23]).…”