Invariance of spectrum for representations of C*-algebras on Banach spaces

Let K be a Banach space, B a unital C*-algebra, and pi : B --> L(K) an injective, unital homomorphism. Suppose that there exists a function gamma : K x K --> R(+) such that, for all k, k(1), k(2) is an element of K, and all b is an element of B, (a) gamma(k, k) = \\k\\(2), (b) gamma(k(1), k(2)) less than or equal to \\K-1\\ \\k(2)\\, (c) gamma(pi(b)k(1), k(2)) = gamma(k(1), pi(b)*k(2)). Then for all b is an element of B, the spectrum of b in B equals the spectrum of pi(b) as a bounded linear operator on K. If gamma satisfies an additional requirement and B is a W*-algebra, then the Taylor spectrum of a commuting n-tuple b = (b(1),...,b(n)) of elements of B equals the Taylor spectrum of the n-tuple pi(b) 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 subset of or equal to K, and bK = {0} only if b = 0, then for each b is an element of 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.