## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 3, Number 2 (2009), 125-141.

### Compact failure of multiplicativity for linear maps between Banach algebras

#### Abstract

We introduce notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a normed space, and prove some general results about these notions. We then consider linear maps $T:A\rightarrow B$ between Banach algebras that are ``close to multiplicative'' in the following senses: the failure of multiplicativity, defined by $S_T(a,b)=T(a)T(b)-T(ab)$ $(a,b\in A)$, is compact [respectively weakly compact]. We call such maps cf-homomorphisms [respectively wcf-homomorphisms]. We also introduce a number of other, related definitions. We state and prove some general theorems about these maps when they are bounded, showing that they form categories and are closed under inversion of mappings and we give a variety of examples. We then turn our attention to commutative $C^*$-algebras and show that the behaviour of the various types of ``close-to-multiplicative'' maps depends on the existence of isolated points in the maximal ideal space. Finally, we look at the splitting of Banach extensions when considered in the category of Banach algebras with bounded cf-homomorphisms [respectively wcf-homomorphisms] as the arrows. This relates to the (weak) compactness of 2-cocycles in the Hochschild-Kamowitz cohomology complex. We prove ``compact'' analogues of a number of established results in the Hochschild-Kamowitz cohomology theory.

#### Article information

**Source**

Banach J. Math. Anal., Volume 3, Number 2 (2009), 125-141.

**Dates**

First available in Project Euclid: 17 December 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1261086716

**Digital Object Identifier**

doi:10.15352/bjma/1261086716

**Mathematical Reviews number (MathSciNet)**

MR2661120

**Zentralblatt MATH identifier**

1201.47038

**Subjects**

Primary: 46H05: General theory of topological algebras

Secondary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25] 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)

**Keywords**

Banach algebra compact multilinear map Banach extension

#### Citation

Heath, Matthew J. Compact failure of multiplicativity for linear maps between Banach algebras. Banach J. Math. Anal. 3 (2009), no. 2, 125--141. doi:10.15352/bjma/1261086716. https://projecteuclid.org/euclid.bjma/1261086716