nLab
Banach algebra

Banach algebras

Definitions

An associative unital Banach algebra is monoid object in the closed monoidal category of Banach spaces (with short linear operators as morphisms, and the usual internal hom, or equivalently the projective tensor product). However, Banach algebras are not usually assumed to be unital, making them semigroup objects (or even magma objects if not assumed to be associative).

Explicitly, this means a Banach space AA equipped with a bilinear multiplication map

m:A×AA, m\colon A \times A \to A ,

which again is usually taken to be associative (and may even be unital), such that

abab, {\|a \cdot b\|} \leq {\|a\|} \cdot {\|b\|} ,

where aba \cdot b (or just aba b) means m(a,b)m(a, b).

In the unital case, we should also require 11{\|1\|} \leq 1, although some authors leave this out. Other authors require 1=1{\|1\|} = 1, which is too strong, since it rules out the trivial algebra. (However, 1=1{\|1\|} = 1 follows from 11{\|1\|} \leq 1 and the existence of any element a0a \ne 0). One can of course always formally adjoin a unit ee with e=1{\|e\|} = 1, forming the Banach algebra AeA \oplus \langle{e}\rangle (using the l 1l^1-direct sum).

The explicit description in terms of mm is of course earlier; but the abstract description as an internal monoid makes clear the most natural definition of Banach coalgebra: a comonoid in the same monoidal category.

YC: An earlier version of this entry said that “the correct” definition of Banach coalgebra is as a comonoid in the usual monoidal category of Banach spaces and short linear operators. I would prefer that this be amended, with similar wording as what I’ve chosen, since experience has shown that the most fruitful candidates for “Banach spaces with coalgebraic structure” are NOT comonoids in this sense. One should instead only require a comultiplication which takes values in something like the injective tensor product, or if you are working with Cstar objects, in something like the spatial tensor product. Does anyone object to my rewording?

Examples

An example of a ‘nonunital’ Banach algebra that has an identity element

Let C nC_n be a cyclic group of order n2n\geq 2 and look at the Banach algebra (in the “strict” sense of a monoid object in BanBan) that is obtained by equipping the Banach space 1(C n)\ell^1(C_n) with the natural convolution product: δ x*δ y=δ x+y\delta_x * \delta_y = \delta_{x+y}. There is a “short” homomorphism from 1(C n)\ell^1(C_n) into the ground field which is just the unique linear extension of the group homomorphism C n{1}C_n \to \{1\} (by the free property of the 1\ell^1-functor) and we let JJ be the kernel of this homomorphism. (JJ is the so-called “augmentation ideal”.) Now JJ is a semigroup object in BanBan and as an algebra it has an identity element pp, but a calculation/hindsight shows that δ ep\delta_e-p must be the constant function C n{1/n}C_n \to \{1/n\}, so that pp has norm (11/n)+(n1)/n=22/n(1-1/n)+(n-1)/n = 2-2/n.

Arens products

If AA is a Banach algebra, its bidual A **A^{**} has two naturally induced Banach algebra structures on it: these are the so-called Arens products on the second dual. These correspond to the left and right tensorial strengths for the bidual monad on the category of Banach spaces (whether with short linear operators as morphisms, or all bounded linear operators). In different language, the two Arens multiplications arise from natural transformations

α AB:A **B **(AB) **\alpha_{A B}: A^{\ast\ast} \otimes B^{\ast\ast} \to (A \otimes B)^{\ast\ast}
\,
β AB:A **B **(AB) **\beta_{A B}: A^{\ast\ast} \otimes B^{\ast\ast} \to (A \otimes B)^{\ast\ast}

described at monoidal monad; putting A=BA = B and post-composing with m **:(AA) **A **m^{\ast\ast}: (A \otimes A)^{\ast\ast} \to A^{\ast\ast} produces the two Arens products. They are named for Richard Arens, who has a 1955 paper which studies this construction in a more general setting . One can see Arens’s “phyla” – with hindsight and Whig history – as a precursor of symmetric closed monoidal categories.

Link to talk by F. E. J. Linton on Arens products

Arens regularity

Algebras where the two Arens products coincide are said to be Arens regular: since B(H)B(H), the algebra of bounded linear operators on a Hilbert space HH, has this property, so do all its closed subalgebras, in particular all C *C^*-algebras. In contrast, if GG is an infinite locally compact group, a result of N. J. Young shows that L 1(G)L^1(G) is not Arens regular. Consequently, it is not isomorphic as a topological algebra to any closed subalgebra of B(H)B(H). Note that this consequence is significantly deeper than the observation that the norm on L 1(G)L^1(G) does not satisfy the C-star identity with respect to the usual involution on L 1(G)L^1(G). See also comments on this MO question.

In general, a Banach algebra AA is Arens regular if and only if, for each μA *\mu\in A^*, the orbit maps aaμa\mapsto a\cdot\mu and aμaa\mapsto \mu\cdot a are weakly compact? as linear maps AA *A\to A^*. (Going from memory here, this result is due to J. S. Pym.) By using known results for factorization of weakly compact? maps, it follows that if AA is Arens regular, one can construct a reflexive Banach space EE and an injective homomorphism of topological algebras AB(E)A\to B(E) which has closed range; this appears to have first been done explicitly by S. Kaijser.

I suspect that here I mean something like a strict monomorphism in the category of monoid objects in TVS, but I am not sure of the details.

However, this is not a characterization of Arens regularity; Young also observed that for any locally compact group GG one may build a reflexive Banach space EE and construct an injective homomorphism of topological algebras L 1(G)B(E)L^1(G)\to B(E) which has closed range.

References

  • Zbigniew Semadeni, Banach spaces of continuous functions, vol. I, gBooks

  • N. Landsman, Mathematical topics between classical and quantum mechanics, Springer

  • Walter Rudin, Functional analysis

  • Richard V. Kadison, John R. Ringrose, Fundamentals of the theory of operator algebras

  • F. F. Bonsall, J. Duncan, Complete normed algebras

  • T. W. Palmer, Banach Algebras and the General Theory of * -Algebras

category: analysis

Revised on November 14, 2013 01:52:39 by Toby Bartels (75.88.43.66)