##
**Perturbations of Banach algebras.**
*(English)*
Zbl 0557.46029

Lecture Notes in Mathematics. 1120. Berlin etc.: Springer-Verlag. V, 118 p. (1985).

There are several sources of the considered theory. One is the classical Banach-Stone theorem stating that the spaces C(S) and C(S’) are isometric if and only if the compacts S and S’ are homeomorphic. Important are also the following generalizations of this result: Uniform algebras A and B are isomorphic if and only if they are isometric (Nagasawa); If the Banach-Mazur distance between the spaces C(S) and C(S’) is smaller then two, then these spaces are isometric (Amir, Cambern); If S and S’ are compact metric spaces then any isometric embedding of C(S) into C(S’) is induced by a continuous map from a subset of S’ onto S (Holsztyński). The ”perturbation problems” mentioned in the introduction are

(i) Let A and B be Banach algebras, both contained in a Banach algebra C. Assume that they are close in a geometric sense (distance between their unit balls is small). Does it follow that A and B are isomorphic as algebras, or at least have some common algebraic properties?

(ii) Assume that we have defined two multiplications xy and \(x\times y\) on a Banach space A, which are close as bilinear maps. What common algebraic properties of (A,\(\cdot)\) and (A,\(\times)\) must appear?

The book gives an account on these and related problems. It is based upon the work of several writers (one should mention B. E. Johnson and R. Rochberg) and a substantial part of its material is due to the author. It is rather a technical book intended for specialists. The basic concept of the book is the concept of an \(\epsilon\)-perturbation of a Banach algebra A, i.e. another continuous multiplication \(\epsilon\)-close in a prescribed sense to the original multiplication of A.

In the first chapter the author presents several concepts of an \(\epsilon\)-perturbation and studies their properties. We quote two corollaries to the main theorem. If the Shilov and Choquet boundaries coincide for a uniform algebra A, then for a sufficiently small perturbation \(\times\) of A these boundaries for (A,\(\times)\) coincide too. For any compact Hausdorff space S the algebra C(S) is stable, i.e. for each sufficiently small perturbation \(\times\) of this algebra the algebra (C(S),\(\times)\) is isomorphic to C(S). The second chapter deals with rather technical results related to mentioned above theorems of Amir-Cambern and Holsztyńnski. In chapter III the author extends Nagasawa theorem to a large class of commutative Banach algebras called here ”natural algebras”, this class contains semisimple commutative unital Banach algebras. In chapter IV there are given properties of a Banach space X ensuring that every isometry (\(\epsilon\)-isometry) from L(X) onto itself is also an algebra isomorphism (\(\epsilon\)-isomorphism).

The last chapter V deals with the crucial concept of the book - the stability. The author considers here only uniform algebras. Put for such algebras A and b \(D(A,B)=\inf \{\epsilon:\) B is algebraically isomorphic to an \(\epsilon\)-perturbation of \(A\}\). A property P of uniform algebras is stable if whenever A has this property, then there is a positive \(\epsilon\) such that \(D(A,B)<\epsilon\) implies that B has the property P too. The following properties are stable: A is Dirichlet, A is antisymmetric, \({\mathfrak M}(A)\) has exactly n components. There is also a result concerning the stability of Shilov-Bishop decomposition of a uniform algebra.

For the rest of the chapter we quote only the titles of sections: § 18 Deformations of algebras of functions on riemann surfaces, § 19 The Hochschild cohomology group and small perturbations, § 20 Perturbation of topological algebras (here by a topological algebra the author means a multiplicatively-convex algebra. The chapter is closed with a list of 20 unsolved problems.

(i) Let A and B be Banach algebras, both contained in a Banach algebra C. Assume that they are close in a geometric sense (distance between their unit balls is small). Does it follow that A and B are isomorphic as algebras, or at least have some common algebraic properties?

(ii) Assume that we have defined two multiplications xy and \(x\times y\) on a Banach space A, which are close as bilinear maps. What common algebraic properties of (A,\(\cdot)\) and (A,\(\times)\) must appear?

The book gives an account on these and related problems. It is based upon the work of several writers (one should mention B. E. Johnson and R. Rochberg) and a substantial part of its material is due to the author. It is rather a technical book intended for specialists. The basic concept of the book is the concept of an \(\epsilon\)-perturbation of a Banach algebra A, i.e. another continuous multiplication \(\epsilon\)-close in a prescribed sense to the original multiplication of A.

In the first chapter the author presents several concepts of an \(\epsilon\)-perturbation and studies their properties. We quote two corollaries to the main theorem. If the Shilov and Choquet boundaries coincide for a uniform algebra A, then for a sufficiently small perturbation \(\times\) of A these boundaries for (A,\(\times)\) coincide too. For any compact Hausdorff space S the algebra C(S) is stable, i.e. for each sufficiently small perturbation \(\times\) of this algebra the algebra (C(S),\(\times)\) is isomorphic to C(S). The second chapter deals with rather technical results related to mentioned above theorems of Amir-Cambern and Holsztyńnski. In chapter III the author extends Nagasawa theorem to a large class of commutative Banach algebras called here ”natural algebras”, this class contains semisimple commutative unital Banach algebras. In chapter IV there are given properties of a Banach space X ensuring that every isometry (\(\epsilon\)-isometry) from L(X) onto itself is also an algebra isomorphism (\(\epsilon\)-isomorphism).

The last chapter V deals with the crucial concept of the book - the stability. The author considers here only uniform algebras. Put for such algebras A and b \(D(A,B)=\inf \{\epsilon:\) B is algebraically isomorphic to an \(\epsilon\)-perturbation of \(A\}\). A property P of uniform algebras is stable if whenever A has this property, then there is a positive \(\epsilon\) such that \(D(A,B)<\epsilon\) implies that B has the property P too. The following properties are stable: A is Dirichlet, A is antisymmetric, \({\mathfrak M}(A)\) has exactly n components. There is also a result concerning the stability of Shilov-Bishop decomposition of a uniform algebra.

For the rest of the chapter we quote only the titles of sections: § 18 Deformations of algebras of functions on riemann surfaces, § 19 The Hochschild cohomology group and small perturbations, § 20 Perturbation of topological algebras (here by a topological algebra the author means a multiplicatively-convex algebra. The chapter is closed with a list of 20 unsolved problems.

Reviewer: W.Żelazko

### MSC:

46J10 | Banach algebras of continuous functions, function algebras |

46H05 | General theory of topological algebras |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |