Multiplicatively spectrum-preserving and norm-preserving maps between invertible groups of commutative Banach algebras

Let $A$ and $B$ be unital semisimple commutative Banach algebras and $T$ a map from the invertible group $A^{-1}$ onto $B^{-1}$. Linearity and multiplicativity of the map are not assumed. We consider the hypotheses on $T$: (1) $\sigma (TfTg)=\sigma (fg)$; (2) $\sigma_{\pi}(TfTg-\alpha)\cap \sigma_{\pi}(fg-\alpha)\ne \emptyset$; (3) $\mathrm{r} (TfTg-\alpha )=\mathrm{r}(fg-\alpha)$ hold for some non-zero complex number $\alpha$ and for every $f, g\in A^{-1}$, where $\sigma (\cdot)$ (resp. $\sigma_{\pi}(\cdot)$) denotes the (resp. peripheral) spectrum and $\rr(\cdot)$ denotes the spectral radius. Under each of the hypotheses we show representations for $T$ and under additional assumptions we show that $T$ is extended to an algebra isomorphism. In particular, if $T$ is a surjective group homomorphism such that $T$ preserves the spectrum or $T$ is a surjective isometry with respect to the spectral radius, then $T$ is extended to an algebra isomorphism. Similar results holds for maps from $A$ onto $B$.