Triangle inequalities in operator algebras

Interpolation and Φ-moment inequalities of noncommutative martingales

“…, x n in L (M). Define T as in the proof of Lemma 4.3 (1). By the noncommutative Khintchine's inequality [24] (also see [22]) and the fact that ( 2 )) such that a i j 's are hermitian operators and a i j = a ji .)…”

Perturbation Formulas for Traces on $C^*$-algebras

“…E A is Murray-von Neumann equivalent to a subprojection of F A , cf. [1]. But this is immediate from Theorem 2.4, because l A o/ is an increasing function, whence for every semi-finite, normal trace T on s/ 9 and thus E A < F A .…”

“…we find /PCS) to be the Hilbert-Schmidt operator with product kernel / [1] (T)fc, where / [1] (T) e C(I x /), given by r f'(x) for x = y…”

“…The Lowner matrix -essentially due to K. Lowner -was used in [7] to give a streamlined version of the theory of operator monotone and operator convex functions. The key observation, [7, 3.4] is that a function / is operator monotone on an interval /, if and only if / [1] (T) is a positive definite matrix for every self-adjoint matrix T (of arbitrary order) with spectrum in J. For the proof of this result we needed a lemma, [7, 3.3] which is the finite-dimensional version of Theorem 2.1 -but phrased in the terminology of Theorem 3.2.…”

On the C*-valued triangle equality and inequality in Hilbert C*-modules

“…It was discovered by Thompson [10] that for every two n-squared complex matrices a and b it holds |a + b| u|a|u * + v|b|v * for appropriately chosen unitary matrices u and v, with the case of equality analyzed in [11]. Thompson's matrix-valued triangle inequality was extended to elements of certain Hilbert modules in [6], while the C * -version of this result was discussed in [1,Theorem 4.2]. In Theorem 2.8 we establish some kind of C * -valued triangle inequality in Hilbert C * -modules.…”