Invariance of spectrum for representations of $\mathbf C^*$-algebras on Banach spaces

Let $\mathcal K$ be a Banach space, $\mathcal B$ a unital $\mathrm C^*$-algebra, and $\pi :\mathcal B \to\mathcal L (\mathcal K )$ an injective, unital homomorphism. Suppose that there exists a function $\gamma :\mathcal K \times \mathcal K\to \mathbb R^+$ such that, for all $k,k_1,k_2\in \mathcal K$, and all $b\in \mathcal B$,

(a) $\gamma (k,k)=\|k\|^2$,

(b) $\gamma (k_1,k_2)\le \|k_1\|\,\|k_2\|$,

(c) $\gamma (\pi _bk_1,k_2)=\gamma (k_1,\pi _{b^*}k_2)$.
Then for all $b\in \mathcal B$, the spectrum of $b$ in $\mathcal B$ equals the spectrum of $\pi _b$ as a bounded linear operator on $\mathcal K$. If $\gamma$ satisfies an additional requirement and $\mathcal B$ is a $\mathrm W ^*$-algebra, then the Taylor spectrum of a commuting $n$-tuple $b=(b_1,\dotsc ,b_n)$ of elements of $\mathcal B$ equals the Taylor spectrum of the $n$-tuple $\pi _b$ in the algebra of bounded operators on $\mathcal K$. Special cases of these results are (i) if $\mathcal K$ is a closed subspace of a unital $\mathrm C^*$-algebra which contains $\mathcal B$ as a unital $\mathrm C ^*$-subalgebra such that $\mathcal {BK}\subseteq \mathcal K$, and $b\mathcal K =\{0\}$ only if $b=0$, then for each $b\in \mathcal B$, the spectrum of $b$ in $\mathcal B$ is the same as the spectrum of left multiplication by $b$ on $\mathcal K$; (ii) if $\mathcal A$ is a unital $\mathrm C ^*$-algebra and $\mathcal J$ is an essential closed left ideal in $\mathcal A$, then an element $a$ of $\mathcal A$ is invertible if and only if left multiplication by $a$ on $\mathcal J$ is bijective; and (iii) if $\mathcal A$ is a $\mathrm C ^*$-algebra, $\mathcal E$ is a Hilbert $\mathcal A$-module, and $T$ is an adjointable module map on $\mathcal E$, then the spectrum of $T$ in the $\mathrm C ^*$-algebra of adjointable operators on $\mathcal E$ is the same as the spectrum of $T$ as a bounded operator on $\mathcal E$. If the algebra of adjointable operators on $\mathcal E$ is a $\mathrm W ^*$-algebra, then the Taylor spectrum of a commuting $n$-tuple of adjointable operators on $\mathcal E$ is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on $\mathcal E$.