Jordan Automorphisms on a Semisimple Banach Algebra

In the group of (continuous) Jordan automorphisms, with the uniform topology, on a semisimple Banach algebra, we show that the connected component of the identity consists of automorphisms. P. Civin and B. Yood have shown that a Jordan homomorphism (that is, a homomorphism that preserves the product xoy = | (xy+yx)) from a Banach algebra onto a semisimple Banach algebra is continuous provided the range algebra satisfies certain conditions [l, Theorem 4.7, p. 783]. These conditions may be reduced to the assumption that the range algebra is semisimple. We sketch the proof and note its similarity to the proof of [4, Theorem 2 ]. Let 6 be a Jordan homomorphism from a Banach algebra A onto a semisimple Banach algebra B. Let x„ tend to zero in A and 6(xn) tend to y in B. Let tt be an algebraically irreducible representation of B on a Banach space. Then ird is a Jordan homomorphism from A onto the primitive algebra tr(B). By a theorem of I. N. Herstein [3, Theorem H, p. 340] irQ is a homomorphism or an antihomomorphism. If wd is an antihomomorphism, we consider A with the reverse product. Therefore we may assume that wd is a homomorphism of a Banach algebra onto the primitive algebra ir(B), and is thus continuous from A into the Banach algebra of bounded linear operators on the representation space of 7r by [4, Theorem l]. This implies that ir(y)=0. We now obtain the continuity of 6 from the semisimplicity of B and the closed graph theorem. The Jordan automorphisms therefore form a topological group in the uniform topology as operators on the algebra with the composition of maps as multiplication. We are concerned with the connected component containing the identity in this topological group. We require the following lemma which is the analog for Jordan automorphisms and derivations of a theorem of G. Zeller-Meier for automorphisms and derivations [7, Theoreme, p. 1131]. 1. Lemma. Let A be a Banach algebra and let a be a continuous Jordan automorphism on A. If the spectrum of a is contained in the open right Received by the editors September 29, 1969. AMS Subject Classifications. Primary 4650, 1740.