The Reduced Minimum Modulus in C^*-Algebras

Abstract.

We define the reduced minimum modulus $\gamma _{\mathcal {A}} (a)$ of a nonzero element a in a unital C ^*-algebra ${\mathcal{A}}$ by $\gamma _{\mathcal {A}} (a) = \rm{inf}\{\parallel a-b \parallel | \rm{AL}({\it a})\subsetneqq AL({\it b}), b \in {\mathcal {A}}\}$. We prove that $\gamma _{\mathcal {A}} (a) = \rm{inf}\{\lambda | \lambda \in \sigma((a^{*}a)^{\frac{1}{2}})\setminus \{0\}\}$. Applying this result to ${\mathcal{A}}$ and its closed two side ideal ${\mathcal{I}}$, we get that dist $(a, \Phi^{c}_{l}({\mathcal {A}})) = \rm{min}\{\lambda | \lambda \in \sigma(\pi((a^{*}a)^{\frac{1}{2}}))\}$, $\forall a \in {\mathcal {A}}\setminus \{0\}$ and $\gamma _{\mathcal {B}}(\pi(a)) = \text{sup}\{\gamma _{\mathcal {A}}(a + k) | k \in {\mathcal {I}}\}$ for any $a \in {\mathcal{A}}\setminus{\mathcal{I}}$ if RR$({\mathcal{A}})$ = 0, where ${\mathcal{B}} = {\mathcal{A}}/{\mathcal{I}}$ and $\pi : {\mathcal{A}} \rightarrow {\mathcal{B}}$ is the quotient homomorphism and $\Phi^{c}_{l} ({\mathcal{A}}) =\{a \in {\mathcal {A}} | \pi(a) \text{is not left invertible in}\,{\mathcal {B}}\}$. These results generalize corresponding results in Hilbert spaces.