A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Note. <s> Well known results of Banach [1]2 and M. H. Stone [8] determine all linear isometric maps of one C(X) onto another (where 'C(X)' denotes, throughout this paper, the set of all real-valued, continuous functions on the compact Hausdorff space X). Such isometries are the maps induced by homeomorphisms of the spaces involved followed by possible changes of sign in the function values on the various closed and open sets. An internal characterization of these isometries would classify them as an algebra isomorphism of the C(X)'s followed by a real unitary multiplication, i.e., multiplication by a real continuous function whose absolute value is 1. The situation in the case of the ring of complex continuous functions (which we denote by 'C'(X)' throughout) is exactly the same; the real unitary multiplication being replaced, of course, by a complex unitary multiplication. It is the purpose of this paper to present the non-commutative extension of the results stated above. A comment as to why this noncommutative extension takes form in a statement about algebras of operators on a Hilbert space seems to be in order. The work of Gelfand-Neumark [2T has as a very particular consequence the fact that each C'(X) is faithfully representable as a self-adjoint, uniformly closed algebra of operators (C*algebra) on a Hilbert space. The representing algebra of operators is, of course, commutative. A statement about the norm and algebraic structure of C' (X) finds then its natural non-commutative extension in the corresponding statement about not necessarily commutative C*algebras. A cursory examination shows that one cannot hope for a word for word transference of the C'(X) result to the non-commutative situation. An isometry between operator algebras is as likely to be an anti-isomorphism as an isomorphism. The direct sum of two C* algebras, which is again a C* algebra, by [2], with an automorphism in one component and an anti-automorphism in the other shows that isomorphisms and anti-isomorphisms together do not encompass all isometries. It is slightly surprising, in view of these facts, that any orderly classification of the isometries of a C* algebra is at all possible. It turns out, in fact, that all isometric maps are composites of a unitary multiplication and a map preserving the C*or quantum mechanical structure (see Segal [7])of the operator algebra in question. More specifically, such maps are linear isomorphisms which commute with the * operation and are multiplicative on powers, composed with a multiplication by a unitary operator in the algebra. <s> BIB001 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Note. <s> We generalize some facts about function algebras to operator algebras, using the noncommutative Shilov boundary or C*-envelope first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are provided. <s> BIB002 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Note. <s> Let : M! N be a zero-product preserving linear map between algebras. We show that under some mild conditions is a product of a central element and an algebra homomorphism. Our result applies to matrix algebras, standard operator algebras, C -algebras and W -algebras. 1. Introduction. Let M and N be algebras over a eld F and : M! N a linear map. We say that is a zero-product preserving map if (a) (b) = 0 in N whenever ab = 0 in M. For example, if h is an element in the center of N and ' : M ! N is an algebra homomorphism then = h' is zero- product preserving. In this paper, we show that in many interesting cases <s> BIB003 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Note. <s> Let A and B be C*-algebras and let T be a linear isometry from A into B. We show that there is a largest projection p in B∗∗ such that T (·)p : A −→ B∗∗ is a Jordan triple homomorphism and T (ab∗c + cb∗a)p = T (a)T (b)∗T (c)p + T (c)T (b)∗T (a)p for all a, b, c in A. When A is abelian, we have ‖T (a)p‖ = ‖a‖ for all a in A. It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces locally to a Jordan triple isomorphism, by a projection. <s> BIB004
Some results similar to Theorem 4.1 and its corollaries have been obtained for operator algebras, W * -algebras and C * -algebras. We refer to BIB002 BIB003 BIB004 for details. The characterizations of isometrics between function spaces and function algebras of integrable functions and analytic functions have also been obtained in terms of the weighted substitution operators. In 1958, Lamperti  proved that every surjective linear isometry on L p (X,µ)(p = 2) is a weighted substitution operator and this result was extended in 1978 by Sourour  to some L p -spaces of vector-valued functions. Linear surjective isometries on Hardy spaces H p (D), p = 2, were characterized in terms of the weighted substitution operators and were extended to Hardy spaces of some vectorvalued analytic functions by Lin and Prolla   . Pathak [20,  studied isometries on spaces of absolutely continuous functions and on C n [0,1]. The isometries of operator algebras and function algebras were also studied. Kadison BIB001 initiated the work on isometries of operator algebras. Recently Blecher and Labuschagne BIB002 have made further investigations regarding isometries on operator algebras. Matheson  has characterized linear isometries between uniform algebras in terms of (generalized) weighted substitution operators. Isometries between nice function spaces are the weighted substitution operators. But we have seen examples of isometries and lattice homomorphisms which cannot be represented as the weighted substitution operators. In characterization of isometries between function spaces, a three-layer mathematical structure is employed, namely, underlying spaces, function spaces over underlying spaces, and dual spaces of function spaces. In most of the cases, evaluation functionals play important role in obtaining a map between underlying spaces giving rise to substitution operator between function spaces. In case of the Banach algebras of functions, the maximal ideal space plays a crucial part. In case it is complicated, the representation of isometric isomorphisms becomes difficult. The surjective isometries, lattice homomorphisms, and disjointness preserving operators between function spaces are important transformations preserving some mathematical structures and it is nice to know that in most of the cases they evolve from two basic binary operations on functions, namely, multiplication and composition justifying the significance of the study of the weighted substitution operators on function spaces.