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Journal of Lie Theory 29 (2019), No. 4, 1045--1070
Copyright Heldermann Verlag 2019

Holomorphic Functions of Exponential Type on Connected Complex Lie Groups

Oleg Yu. Aristov
Obninsk 249038, Russia

Holomorphic functions of exponential type on a complex Lie group G (introduced by Akbarov) form a locally convex algebra, which is denoted by Oexp(G). Our aim is to describe the structure of Oexp(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 Oexp(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 Oexp(G) is the algebra of all holomorphic functions.

Keywords: Complex Lie group, linear group, holomorphic function of exponential type, Arens-Michael envelope, submultiplicative weight, length function, exponential radical.

MSC: 22E10, 22E30, 32A38, 46F05.

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