Holomorphic functions of exponential type on connected complex Lie groups

Holomorphic functions of exponential type on a complex Lie group G (introduced by Akbarov) form a locally convex algebra, which is denoted by O_exp(G). Our aim is to describe the structure of O_exp(G) in the case when G is connected. The following topics are auxiliary for the claimed purpose but of independent interest: (1) a characterization of linear complex Lie group (a result similar to that of Luminet and Valette for real Lie groups); (2) properties of the exponential radical when G is linear; (3) an asymptotic decomposition of a word length function into a sum of three summands (again for linear groups). The main result presents O_exp(G) as a complete projective tensor of three factors, corresponding to the length function decomposition. As an application, it is shown that if G is linear then the Arens-Michael envelope of O_exp(G) is the algebra of all holomorphic functions.