Invariance of Spectrum for Representations of C∗-algebras on Banach Spaces

Let K be a Banach space, B a unital C∗-algebra, and π : B → L(K) an injective, unital homomorphism. Suppose that there exists a function γ : K×K → R+ such that, for all k, k1, k2 ∈ K, and all b ∈ B, (a) γ(k, k) = ‖k‖2, (b) γ(k1, k2) ≤ ‖k1‖ ‖k2‖, (c) γ(πbk1, k2) = γ(k1, πb∗k2). Then for all b ∈ B, the spectrum of b in B equals the spectrum of πb as a bounded linear operator on K. If γ satisfies an additional requirement and B is a W∗-algebra, then the Taylor spectrum of a commuting n-tuple b = (b1, . . . , bn) of elements of B equals the Taylor spectrum of the n-tuple πb in the algebra of bounded operators on K. Special cases of these results are (i) if K is a closed subspace of a unital C∗-algebra which contains B as a unital C∗-subalgebra such that BK ⊆ K, and bK = {0} only if b = 0, then for each b ∈ B, the spectrum of b in B is the same as the spectrum of left multiplication by b on K; (ii) if A is a unital C∗-algebra and J is an essential closed left ideal in A, then an element a of A is invertible if and only if left multiplication by a on J is bijective; and (iii) if A is a C∗-algebra, E is a Hilbert A-module, and T is an adjointable module map on E, then the spectrum of T in the C∗-algebra of adjointable operators on E is the same as the spectrum of T as a bounded operator on E. If the algebra of adjointable operators on E is a W∗-algebra, then the Taylor spectrum of a commuting n-tuple of adjointable operators on E is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on E.