Derivations and automorphisms of Banach algebras of power series

The group of automorphisms of $L\sp 1(0,1)$ is connected

“…In [1] we have shown that if an automorphism 6 of V is extended to M[0,1 ), then there exists a complex number z, such that for every x G [0,1 ), (1) e(âx) = ezxôx + px> where a(px) > x and px({x}) = 0 (for every measure p., we denote the infimum of the support of p. by a(p.) ). Following the terminology used by S. Grabiner [2] for the automorphisms of the power series algebra we call an automorphism 6 of V principal if z = 0 in (1). If an automorphism 6 of V satisfies (1), then (2) e-zde(Sx) = Sx + e-zdftx (xg[0,1)), with a(e~zdpx) > x and (e~zdpx)({x}) = 0.…”

“…Following the terminology used by S. Grabiner [2] for the automorphisms of the power series algebra we call an automorphism 6 of V principal if z = 0 in (1). If an automorphism 6 of V satisfies (1), then (2) e-zde(Sx) = Sx + e-zdftx (xg[0,1)), with a(e~zdpx) > x and (e~zdpx)({x}) = 0. Therefore e~zd6 is a principal automorphism and in order to show that every automorphism is of the form ez eq , it suffices to show that every principal automorphism is in the component of the identity.…”

“…Now if we examine the image of ôx under the two sides of (2) we will see that z = -1 (see the proof of Theorem 1.4 in [2]). Thus (3) Qe-dQ~x = e~deq .…”

Quasicompact endomorphisms of commutative semiprime Banach algebras

“…Examples of commutative semiprime Banach algebras which are not semisimple include certain Banach algebras of formal power series, as discussed in [10]. In particular, A(D) and H p (D) for p ∈ [1, ∞) are commutative radical semiprime Banach algebras with respect to convolution multiplication defined by…”

“…We recall the following terminology and notation from [10]. The algebra of complex formal power series in one variable is denoted by C[[z]].…”