Let $v(x)$ be the numerical radius of an element $x$ in a $C^*$-algebra $\mathfrak{A}$. First, we prove several numerical radius inequalities in $\mathfrak{A}$. Particularly, we present a refinement of the triangle inequality for the numerical radius in $C^*$-algebras. In addition, we show that if $x\in\mathfrak{A}$, then $v(x) = \frac{1}{2}\|x\|$ if and only if $\|x\| = \|\mbox{Re}(e^{iθ}x)\| + \|\mbox{Im}(e^{iθ}x)\|$ for all $θ\in \mathbb{R}$. Among other things, we introduce a new type of parallelism in $C^*$-algebras based on numerical radius. More precisely, we consider elements $x$ and $y$ of $\mathfrak{A}$ which satisfy $v(x + λx) = v(x) + v(y)$ for some complex unit $λ$. We show that this relation can be characterized in terms of pure states acting on $\mathfrak{A}$.
We present a necessary and sufficient condition for the norm-parallelism of bounded linear operators on a Hilbert space. We also give a characterization of the Birkhoff--James orthogonality for Hilbert space operators. Moreover, we discuss the connection between norm-parallelism to the identity operator and an equality condition for the Davis--Wielandt radius. Some other related results are also discussed.
Let $a$ be a positive element in a unital $C^*$-algebra $\mathfrak{A}$. We define a semi-norm on $\mathfrak{A}$, which generalizes the $a$-operator semi-norm and the $a$-numerical radius. We investigate basic properties of this semi-norm and prove inequalities involving it. Further, we derive new upper and lower bounds for the $a$-numerical radii of elements in $\mathfrak{A}$. Some other related results are also discussed.
It is shown that the spectral radius is continuous on a $C^*$-algebra if and only if the $C^*$-algebra is type I. This answers a question of V. Shulman and Yu.~Turovskii [10]. It is shown also that the closure of nilpotents in a $C^*$-algebra contains an element with non-zero spectrum if and only if the $C^*$-algebra is not type I.
Let $X$ be a compact metric space, let $A$ be a unital AH algebra with large matrix sizes, and let $B$ be a stably finite unital C*-algebra. Then we give a lower bound for the radius of comparison of $C(X) \otimes B$ and prove that the dimension-rank ratio satisfies $\operatorname{drr} (A) = \operatorname{drr} \left(C(X)\otimes A\right)$. We also give a class of unital AH algebras $A$ with $\operatorname{rc} \left(C(X) \otimes A\right) = \operatorname{rc} (A)$. We further give a class of stably finite exact $\mathcal{Z}$-stable unital C*-algebras with nonzero radius of comparison.
Let $(X, Γ)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $Γ$ is a discrete amenable group. It is shown that, if $(X, Γ)$ has a version of Rokhlin property (uniform Rokhlin property) and if $\mathrm{C}(X)\rtimesΓ$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes Γ$ is at most half of the mean topological dimension of $(X, Γ)$. These two conditions are shown to be satisfied if $Γ= \mathbb Z$ or if $(X, Γ)$ is an extension of a free Cantor system and $Γ$ has subexponential growth. …
Consider a minimal free topological dynamical system $(X, T, \mathbb{Z}^d)$. It is shown that the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ is at most the half of the mean topological dimension of $(X, T, \mathbb{Z}^d)$. As a consequence, the C*-algebra $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ is classifiable if $(X, T, \mathbb{Z}^d)$ has zero mean dimension.
Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let α\colon G \to Aut (A) be an action of G on A which has the weak tracial Rokhlin property. Let A^α be the fixed point algebra. Then the radius of comparison satisfies rc (A^α) \leq rc (A) and rc ( C* (G, A, α) ) \leq ( 1 / card (G) ) rc (A). The inclusion of A^α in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^α) to the fixed points of the purely positive part of Cu (A), and the purely positive part of Cu ( C* (G, A, α) ) is isomorphic to this semigroup…
