Abstract: 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.
Keywords: a-numerical radius; a-Birkhoff-James orthogonality; a-norm parallelism; a-numerical radius parallelism; C^*-algebra; state space; a-numerical range
DOI: DOI 10.14712/1213-7243.2026.007
AMS Subject Classification: 46L05 47A12 46B20 46C50