A holomorphic functional calculus for finite families of commuting semigroups

Let A be a commutative Banach algebra such that uA = {0} for u $\in$ A \ {0} which possesses dense principal ideals. The purpose of the paper is to give a general framework to define F (--$\lambda$1$\Delta$T 1 ,. .. , --$\lambda$ k $\Delta$T k) where F belongs to a natural class of holomorphic functions defined on suitable open subsets of C k containing the "Arveson spectrum" of (--$\lambda$1$\Delta$T 1 ,. .. , --$\lambda$ k $\Delta$T k), where $\Delta$T 1 ,. .. , $\Delta$T k are the infinitesimal generators of commuting one-parameter semigroups of multipliers on A belonging to one of the following classes (1) The class of strongly continous semigroups T = (T (te ia)t>0 such that $\cup$t>0T (te ia)A is dense in A, where a $\in$ R. (2) The class of semigroups T = (T ($\zeta$)) $\zeta$$\in$S a,b holomorphic on an open sector S a,b such that T ($\zeta$)A is dense in A for some, or equivalently for all $\zeta$ $\in$ S a,b. We use the notion of quasimultiplier, introduced in 1981 by the author at the Long Beach Conference on Banach algebras: the generators of the semigroups under consideration will be defined as quasimultipliers on A, and for $\zeta$ in the Arveson resolvent set $\sigma$ar($\Delta$T) the resolvent ($\Delta$T -- $\zeta$I) --1 will be defined as a regular quasimultiplier on A, i.e. a quasimultiplier S on A such that sup n$\ge$1 $\lambda$ n S n u < +$\infty$ for some $\lambda$ > 0 and some u generating a dense ideal of A and belonging to the intersection of the domains of S n , n $\ge$ 1. The first step consists in "normalizing" the Banach algebra A, i.e. continuously embedding A in a Banach algebra B having the same quasi-multiplier algebra as A but for which lim sup t$\rightarrow$0 + T (te ia) M(B) < +$\infty$ if T belongs to the class (1), and for which lim sup $\zeta$$\rightarrow$0 $\zeta$$\in$S $\alpha$,$\beta$ T ($\zeta$) < +$\infty$ for all pairs ($\alpha$, $\beta$) such that a < $\alpha$ < $\beta$ < b if T belongs to the class (2). Iterating this procedure this allows to consider ($\lambda$j$\Delta$T j + $\zeta$I) --1 as an element of M(B) for $\zeta$ $\in$ Resar(--$\lambda$j$\Delta$T j), the "Arveson resolvent set " of --$\lambda$j$\Delta$T j , and to use the standard integral 'resolvent formula' even if the given semigroups are not bounded near the origin. A first approach to the functional calculus involves the dual G a,b of an algebra of fast decreasing functions, described in Appendix 2. Let a = (a1,. .. , a k), b = (b1,. .. , b k), with aj $\le$ bj $\le$ aj + $\pi$, and denote by M a,b the set of families ($\alpha$, $\beta$) = ($\alpha$1, $\beta$1),. .. , ($\alpha$ k , $\beta$ k) such that 1