Abstract. We present an operator space version of Rieffel's theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As an application we obtain a necessary and sufficient condition for the matrix metric from an unbounded Fredholm module to give the BW-topology on the matrix state space of the C * -algebra. Motivated by recent results we formulate a noncommutative Lipschitz seminorm on a matrix order unit space and characterize those matrix Lipschitz seminorms whose matrix metric topology coincides with the BW-topology on the matrix state space.

A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum GromovHausdorff distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov-Hausdorff distance is zero. We establish a completeness theorem. As applications, we show that a quantized metric space with 1-exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized Gromov-Hausdorff distance, and that matrix algebras converge naturally to the sphere for quantized Gromov-Hausdorff distance.

Abstract. In this paper, we investigate the problem of when a C * -algebra is commutative through operator-monotonic increasing functions. The principal result is that the function e t , t ∈ [0, ∞), is operator-monotonic increasing on a C * -algebra A if and only if A is commutative. Therefore, C * -algebra A is commutative if and only if e x+y = e x e y in A+C for all positive elements x, y in A.

In this paper we consider system of inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations. A Newton-type algorithm is applied to solve iteratively the smooth equations so that a solution of the problem concerned is found. We show that the algorithm is globally and locally quadratically convergent under suitable assumptions. Preliminary numerical results are reported.

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