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We will extend the main result of [1] to the non-unital case with a totally different proof. More precisely, we give an abstract characterization of an arbitrary self-adjoint weak⁎-closed subspace of L (H) (equipped with the induced matrix norm, the induced matrix cone and the induced weak⁎-topology). In order to do this, we give a matrix analogue of a result of Bonsall for ⁎-operator spaces equipped with closed matrix cones. [ABSTRACT FROM AUTHOR]

As a non-unital analogue of the main result in [6] , we show in this article that if A is a separable quasi-central C ⁎ -algebra with property T and is nuclear, then there is a sequence of positive integers { n k } k ∈ N such that the c 0 -direct sum of the family { M n k } k ∈ N of matrix algebras is an ideal J A of A and that A / J A has no tracial state. In particular, a separable C ⁎ -algebra A is a c 0 -direct sum of matrix algebras if and only if A is quasi-central, is nuclear, has property T and every unital simple quotient of A has a tracial state. We also show that a separable property T C ⁎ -algebra A is residually finite dimensional if and only if A admits an amenable faithful tracial state and every ideal of A has non-zero center. [ABSTRACT FROM AUTHOR]

Let 1 ≤ p ≤ + ∞ . We show that the positive part of the closed unit ball of a non-commutative L p -space, as a metric space, is a complete Jordan ⁎ -invariant for the underlying von Neumann algebra. [ABSTRACT FROM AUTHOR]

In this short note, we give a brief study of the *-algebra R ⊗∞ which is given by certain “inductive limit process” of the algebraic tensor product R ⊗n of the C*-algebra R representing quantum mechanics with one particle. In particular, we will study “feasible pure states” of R ⊗∞ and their GNS constructions. [Copyright &y& Elsevier]