Automatic continuity for Banach algebras

The basic question in automatic continuity theory is the following. Let $A$ and $B$ be Banach algebras (cf. Banach algebra), and let $\theta : A \rightarrow B$ be a homomorphism. What algebraic conditions on $A$ and/or $B$ ensure that the homomorphism $\theta$ is automatically continuous? A variation of the question is the following. Let $A$ be a Banach algebra, let $E$ be a Banach $A$-bimodule, and let $D : A \rightarrow E$ be a derivation (cf. also Derivation in a ring). What algebraic conditions on $A$ and/or $E$ ensure that $D$ is automatically continuous? There are important generalizations of the latter question: for various purposes it is important to replace the derivation by the more general notion of an "intertwining mapping" . A special case of the automatic continuity problem for homomorphisms is the uniqueness-of-norm problem, which asks which Banach algebras have a unique complete algebra norm. For a substantial recent (as of 2000) account of automatic continuity theory for Banach algebras, see [a4]; all the terms that are used here are defined in [a4].

The starting point for automatic continuity theory is the easily proved fact that every character on a Banach algebra $A$ (i.e., every homomorphism from $A$ onto the complex field $\mathbf{C}$) is automatically continuous (cf. also Continuous function). This was already stated in the seminal work of I.M. Gel'fand around 1940. Note that there is a deep related question. Let $A$ be a Fréchet algebra (so that the topology of $A$ is given by a sequence of algebra semi-norms on $A$, and $A$ is complete). Then it is an open question (as of 2000) whether or not every character on $A$ is automatically continuous. This is called Michael's problem, because it was raised in [a12]. A positive result is known in many cases; a striking sufficient condition, involving analytic functions of several complex variables, for the continuity of all such characters is given in [a7].

It follows easily from the continuity of characters that every homomorphism $\theta : A \rightarrow B$ from a Banach algebra $A$ into a commutative semi-simple Banach algebra $B$ is continuous. A closely related result is Johnson's uniqueness-of-norm theorem: Every semi-simple Banach algebra has a unique complete algebra norm. For lovely alternative proofs of this theorem, see [a1] and [a13]. There are non-semi-simple commutative Banach algebras having a unique complete algebra norm. For example, this is true of the convolution algebras $L ^ { 1 } ( \mathbf{R} ^ { + } , \omega )$, where $\omega$ is a weight function on $\mathbf{R} ^ { + }$ (see [a4], § 5.2). On the other hand, there are even commutative Banach algebras with a one-dimensional (Jacobson) radical which do not have a unique complete algebra norm (see [a4], § 5.1). Nevertheless there are striking open questions in this area: it is not known (as of 2000) whether a commutative Banach algebra which is an integral domain necessarily has a unique complete algebra norm; the question is also open for Banach algebras with a finite-dimensional radical; for partial results, see [a5]. The following question is also open. Let $A$ and $B$ be Banach algebras, and let $\theta : A \rightarrow B$ be a homomorphism. Suppose that $B$ is semi-simple and that $\overline { \theta ( A ) } = B$. Is $\theta$ automatically continuous?

The separating space $\mathfrak S ( T )$ of a linear mapping $T : E \rightarrow F$, where $E$ and $F$ are Banach spaces, is defined to be the set of elements $y \in F$ such that there is a sequence $( x _ { n } )$ in $E$ with $x _ { n } \rightarrow 0$ in $E$ and $T x _ { n } \rightarrow y$ in $F$. Clearly, $\mathfrak S ( T )$ is a closed linear subspace of $F$ and, by the closed-graph theorem, $\mathfrak { S } ( T ) = \\{ 0 \\}$ if and only if $T$ is continuous. A key result in automatic continuity theory is the stability lemma. One version of this is as follows; there are many variations. Let $E$ and $F$ be Banach spaces, let $T : E \rightarrow F$ be a linear mapping, let $( E _ { n } : n \in {\bf Z} ^ { + } )$ be a sequence of Banach spaces with $E _ { 0 } = E$, and let $R _ { n } \in \mathcal{B} ( E _ { n } , E _ { n - 1 } )$ ($n \in \mathbf N$). Then $( \mathfrak { S } ( T R _ { 1 } \ldots R _ { n} ) : n \in \bf N )$ is a nest in $F$ that stabilizes. This leads to a proof [a11] that all derivations from a semi-simple Banach algebra to itself are automatically continuous, and to many other results.

Let $A$ be a Banach algebra, let $E$ and $F$ be Banach $A$-bimodules, and let $T : E \rightarrow F$ be a linear mapping. The continuity ideal ${\cal I} ( T )$ of $T$ is defined to be $\\{ a \in A : a . \mathfrak{S} ( T ) = \mathfrak{S} ( T ) , a = \\{ 0 \\} \\}$. This is an ideal in $A$, and the "bigger IT is, the more continuous T is" . This leads to proofs that all derivations from various Banach algebras $A$ into each Banach $A$-bimodule are automatically continuous; see [a4], § 5.3. For example, this is true whenever $A$ is a $C ^ { * }$-algebra, [a14].

Let $\theta : A \rightarrow B$ be a homomorphism between Banach algebras. The main boundedness theorem of W.G. Bade and P.C. Curtis Jr. [a2] asserts that, in the case where $A$ has many idempotents, the continuity ideal $\mathcal{I} ( \theta )$ is necessarily "large" . This leads to a proof that every homomorphism from many algebras, including the algebra $\mathcal{B} ( E )$ of all bounded linear operators on a Banach space $E$ for certain Banach spaces $E$, is automatically continuous.

Let $C ( \Omega )$ be the commutative $C ^ { * }$-algebra of all continuous functions on a compact space $\Omega$, taken with the uniform norm on $\Omega$. The theory of Bade and Curtis shows that each homomorphism from $C ( \Omega )$ into a Banach algebra must be continuous on a dense subalgebra of $C ( \Omega )$; they left open the question of whether such a homomorphism is necessarily continuous on the whole of $C ( \Omega )$. Eventually it was proved [a3], [a8] (see [a4], § 5.7) that this is not the case: For each infinite compact space $\Omega$, there is a discontinuous homomorphism from $C ( \Omega )$ into certain commutative Banach algebras. Indeed, it is known just which Banach algebras arise in this situation. Note that the proof of this theorem requires the assumption of the continuum hypothesis CH; that some additional set-theoretic hypothesis is required is a remarkable result that is discussed in [a6]. In fact, there is a discontinuous homomorphism from "most" , perhaps all, infinite-dimensional, commutative Banach algebras. It is an attractive result of J.R. Esterle [a9] that all epimorphisms from $C ( \Omega )$ onto a Banach algebra are automatically continuous; the analogous question for $C ^ { * }$-algebras is open (as of 2000).

Let $G$ be a locally compact group, and let $L ^ { 1 } ( G )$ be the corresponding group algebra. It is an active area of research to determine whether or not all derivations from $L ^ { 1 } ( G )$ into a Banach $L ^ { 1 } ( G )$-bimodule are automatically continuous. That this is true for many such groups $G$ is proved in [a4], §5.6; a key paper on which this work is based is [a15]. It is a challenging open question (as of 2000) to determine whether or not this is true for all groups $G$, or even for all discrete groups, in which case the corresponding group algebra is denoted by ${\bf l}^ { 1 } ( G )$.

#### References[edit]

[a1] |  B. Aupetit, "The uniqueness of complete norm topology in Banach algebras and Banach Jordan algebras" J. Funct. Anal. , 47 (1982) pp. 1–6

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[a2] |  W.G. Bade, P.C. Curtis Jr., "Homomorphisms of commutative Banach algebras" Amer. J. Math. , 82 (1960) pp. 589–608
[a3] |  H.G. Dales, "A discontinuous homomorphism from $C ( X )$" Amer. J. Math. , 101 (1979) pp. 647–734
[a4] |  H.G. Dales, "Banach algebras and automatic continuity" , London Math. Soc. Monographs , 24 , Clarendon Press (2001)
[a5] |  H.G. Dales, R.J. Loy, "Uniqueness of the norm topology for Banach algebras with finite-dimensional radical" Proc. London Math. Soc. (3) , 74 (1997) pp. 633–661
[a6] |  H.G. Dales, W.H. Woodin, "An introduction to independence for analysts" , London Math. Soc. Lecture Notes , 115 , Cambridge Univ. Press (1987)
[a7] |  P.G. Dixon, J.R. Esterle, "Michael's problem and the Poincaré–Bieberbach phenomenon" Bull. Amer. Math. Soc. , 15 (1986) pp. 127–187
[a8] |  J.R. Esterle, "Sur l'existence d'un homomorphisme discontinu de $\mathcal{C} ( K )$" Proc. London Math. Soc. (3) , 36 (1978) pp. 46–58
[a9] |  J.R. Esterle, "Theorems of Gelfand–Mazur type and continuity of epimorphisms from $\mathcal{C} ( K )$" J. Funct. Anal. , 36 (1980) pp. 273–286
[a10] |  B.E. Johnson, "The uniqueness of the (complete) norm topology" Bull. Amer. Math. Soc. , 73 (1967) pp. 537–539
[a11] |  B.E. Johnson, A.M. Sinclair, "Continuity of derivations and a problem of Kaplansky" Amer. J. Math. , 90 (1968) pp. 1067–1073
[a12] |  E.A. Michael, "Locally multiplicatively-convex topological algebras" Memoirs Amer. Math. Soc. , 11 (1952)
[a13] |  T.J. Ransford, "A short proof of Johnson's uniqueness-of-norm theorem" Bull. Amer. Math. Soc. , 21 (1989) pp. 487–488
[a14] |  J.R. Ringrose, "Automatic continuity of derivations of operator algebras" J. London Math. Soc. (2) , 5 (1972) pp. 432–438
[a15] |  G.A. Willis, "The continuity of derivations from group algebras: factorizable and connected groups" J. Austral. Math. Soc. (Ser. A) , 52 (1992) pp. 185–204