Invariance of spectrum for representations of $\mathrm {C}*$-algebras on Banach spaces

Abstract:

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 $\mathcal {E}$.