1987
DOI: 10.1017/s0017089500006704
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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 operato… Show more

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“…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 !!^!!…”
Section: Some Known Results On the Associativity Of Jb*-algebrasmentioning
confidence: 94%
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“…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 !!^!!…”
Section: Some Known Results On the Associativity Of Jb*-algebrasmentioning
confidence: 94%
“…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).…”
Section: Some Known Results On the Associativity Of Jb*-algebrasmentioning
confidence: 99%
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