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Let C(X) be the space of continuous functions on a compact T2-space X where each point of Jf is a Gs. If F: B -» C(X) is a biholomorphic (in the sense that F and F ~ ' are Frechet differentiable) map of B = {/1 || /1| < 1} onto a convex domain with DF(0) = I, then F is Lorch analytic (i.e., DF(fXg) = afg for some ai e C(X)). Let R be a commutative Banach algebra with identity such that the Gelfand homomorphism of R into CC3H) is an isometry. Starlike, convex, close-to-convex, spirallike and $-like functions are defined in B = {x £ * I 11*11 < 1} for ?-analytic functions in B and they are related to associated complex-valued holomorphic functions in A = {z e C| |z| < 1).

Starlike, convex, close-to-convex, spiral-like, and Φ-like maps in a commutative Banach algebra with identity