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Keywords:
$a$-numerical radius; $a$-Birkhoff--James orthogonality; $a$-norm parallelism; $a$-numerical radius parallelism; $C^*$-algebra; state space; $a$-numerical range
$a$-numerical radius; $a$-Birkhoff--James orthogonality; $a$-norm parallelism; $a$-numerical radius parallelism; $C^*$-algebra; state space; $a$-numerical range
Summary:
Let $\mathcal{A}$ be a unital $C^*$-algebra and let $a\in\mathcal{A}$ be a positive and invertible element. Suppose that $\mathcal{S}(\mathcal{A})$ is the set of all states on $\mathcal{\mathcal{A}}$ and let $$ \mathcal{S}_a (\mathcal{A})=\Big\{\frac{f}{f(a)} \colon f \in \mathcal{S}(\mathcal{A}), f(a)\neq 0\Big\}. $$ We introduce a family of generalized norms, called $(a,\lambda)$-norms, on $\mathcal{A}$ defined by $$ \|x\|_{a,\lambda}:=\sup\Big\{\sqrt{\lambda\varphi(x^*ax) +(1-\lambda)|\varphi(ax)|^2} \colon \varphi\in\mathcal{S}_a(\mathcal{A})\Big\},\qquad \lambda\in [0,1]. $$ This family of norms generalizes the recently introduced $a$-operator norm, $\|{\cdot}\|_a$ and $a$-numerical radius norm, $v_a({\cdot})$ in unital $C^*$-algebras. The notions of Birkhoff--James orthogonality and norm-parallelism with respect to $\|{\cdot}\|_{a,\lambda}$, which is called, $(a,\lambda)$-Birkhoff--James orthogonality and $(a,\lambda)$-norm parallelism in $\mathcal{A}$, respectively, are introduced and investigated. Characterizations of $(a,\lambda)$-norm parallelism and $(a,\lambda)$-Birkhoff--James orthogonality in terms of the elements of $\mathcal{S}_a(\mathcal{A})$ are obtained. In particular, the relationship between these new concepts are described. Our results extend and cover some known results in this area.
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