Algebras of polynomials generated by linear operators

Let E be a Banach space and A be a commutative Banach algebra with identity. Let P(E, A) be the space of A-valued polynomials on E generated by bounded linear operators (an n-homogenous polynomial in P(E,A) is of the form P=∑ᵢ₌₁^∞ Tⁿᵢ, where Tᵢ:E→ A, 1≤ i <∞, are bounded linear operators and ∑ᵢ₌₁^∞ \|Tᵢ\|ⁿ < ∞). For a compact set K in E, we let P(K, A) be the closure in C(K,A) of the restrictions P|K of polynomials P in P(E,A). It is proved that P(K, A) is an A-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product PN(K)⊗_ε A, where PN(K) is the uniform algebra on K generated by nuclear scalar-valued polynomials. The character space of P(K, A) is then identified with K̂N× M(A), where KN is the nuclear polynomially convex hull of K in E, and M(A) is the character space of A.