Completely almost periodic functionals
Abstract
Using the notion of complete compactness introduced by H. Saar, we define completely almost periodic functionals on completely contractive Banach algebras. We show that, if is a Hopf–von Neumann algebra with injective, then the space of completely almost periodic functionals on is a subalgebra of .
Keywords: completely compact map; completely almost periodic functional; Hopf–von Neumann algebra. 2000 Mathematics Subject Classification: Primary 47L25; Secondary 22D25, 43A30, 46L07, 47L50.
Introduction
The almost periodic and weakly almost periodic continuous functions on a locally compact group form subalgebras of , usually denoted by and , respectively: this is fairly elementary to prove and well known (see [Bur] for instance). In a more abstract setting, one can define, for a general Banach algebra the spaces and of almost periodic and weakly almost periodic functionals on ; if , we have and .
For , Eymard’s Fourier algebra ([Eym]), the spaces and are usually denoted by and : they were first considered by C. F. Dunkl and D. E. Ramirez ([D–R]) and further studied by E. E. Granirer ([Gra 1] and [Gra 2]), A. T.M. Lau ([Lau]), and others. Except in a few special cases, e.g., if is abelian or discrete and amenable, it is unknown whether and are subalgebras of the group von Neumann algebra .
Recently, M. Daws considered the almost periodic and weakly almost periodic functionals on the predual of a Hopf–von Neumann algebra with underlying von Neumann algebra ([Daw]). He proved: If is abelian, then both and are subalgebras of ([Daw, Theorems 1 and 4]). Unfortunately, the demand that be abelian is crucial for Daws’ proofs to work (see [Run] for a discussion).
Over the past two decades, it has become apparent that purely Banach algebraic notions aren’t well suited for the study of : one often has to tweak these notions in a way that takes the canonical operator space structure of —as the predual of the group von Neumann algebra—into account. For instance, Banach algebraic amenability of forces to be finitebyabelian ([F–R]) whereas is operator amenable if and only if is amenable ([Rua]), a much more satisfactory result.
We apply this philosophy to almost periodicity. A functional on a Banach algebra is called almost periodic if the maps
() 
are compact. Suppose now that is a completely contractive Banach algebra. Then the maps ( ‣ Introduction) are completely bounded. There are various definitions that attempt to fit the notion of a compact operator to a completely bounded context (see [Saa], [Web], [Oik], and [Yew], for instance). We focus on the definition of a completely compact map from [Saa], and define to be completely almost periodic if the maps ( ‣ Introduction) are completely compact.
Our main result is that, if is a Hopf–von Neumann algebra such that is injective, then the completely almost periodic functionals on form a subalgebra of . This applies, in particular, to in the cases where is amenable or connected.
1 Completely compact maps
There are various ways to adapt the notion of a compact operator to an operator space setting: operator compactness ([Web] and [Yew]), complete compactness ([Saa]), and Gelfand complete compactness ([Oik]).
The notion of a completely compact map between algebras was introduced by H. Saar in his Diplomarbeit [Saa] under G. Wittstock’s supervision. The starting point of his definition is the following observation: if and are Banach spaces, and is a bounded linear map, then is compact if and only if, for each , there is a finitedimensional subspace of such that , where is the quotient map.
This can be used to define an operator space analog of compactness, namely complete compactness. Saar didn’t define complete compactness for maps between general, abstract operator spaces—simply because these objects hadn’t been defined yet at that time—, but his definition obviously extends to general operator spaces. (Our reference for operator space theory is [E–R], the notation and terminology of which we adopt.) In modern language, Saar’s definition reads:
Definition 1.1.
Let and be operator spaces. Then is called completely compact if, for each , there is a finitedimensional subspace of such that , where is the quotient map.
Remarks.

Trivially, completely compact maps are compact.

It is obvious that, if is a Banach space and is an operator space, then is completely compact if and only if it is compact.

On [Saa, pp. 32–34], Saar constructs an example of a compact, completely bounded map on that fails to be completely compact.

Complete compactness may not be stable under corestrictions, i.e., if be completely compact, and let be a closed subspace of containing , then it is not clear why viewed as an element of should be completely compact.
Given two operator spaces and , we write for the completely compact operators in .
The following proposition is essentially [Saa, Lemma 1 a) and Lemma 2]. (Of course, Saar only considers maps between algebras, but his proofs carry over more or less verbatim.)
Proposition 1.2.
Let and be operator spaces. Then:

is a closed subspace of containing all finite rank operators;

if , is another operator space, and , then ;

if , is another operator space, and , then .
From Schauder’s theorem and Saar’s characterization of compact operators, it follows immediately that a bounded linear operator from a Banach space into a Banach space is compact if and only if, for each , there is a closed subspace of with finite codimension such that (compare also [Lac]).
Following [Oik], we define:
Definition 1.3.
Let and be operator spaces. Then is called Gelfand completely compact if, for each , there is a closed subspace of with finite codimension such that .
Remarks.

Obviously, is Gelfand completely compact if and only if is completely compact (and vice versa).

A result analogous to Proposition 1.2 holds for Gelfand completely compact maps.
Under certain circumstances, every completely compact map is Gelfand completely compact ([Oik, Theorem 3.1]). For our purposes, the following is important (see [E–R, p. 70] for the notion of an injective operator space):
Proposition 1.4.
Let and be operator spaces such that and are injective. Then the following are equivalent for :

is completely compact;

is Gelfand completely compact;

is a norm limit of finite rank operators.
Proof.
Obviously, (iii) implies both (i) and (ii).
(ii) (iii): Let be Gelfand completely compact, and let . Then there is a closed subspace of with finite codimension such that . Due to the injectivity of , there is such that and . Then satisfies and vanishes on ; since has finite codimension, must have finite rank..
(i) (iii): As is Gelfand completely compact, it is a norm limit of finite rank operators—by the argument used for (ii) (iii)—as is its adjoint . Hence, given , there is a finite rank operator such that . Let be the Dixmier projection, i.e., the adjoint of the canonical embedding of into . Then is a finite rank operator from to such that . ∎
2 Completely almost periodic functionals
If is a Banach algebra, then its dual spaces becomes a Banach bimodule via
For , define via
A functional is commonly called almost periodic if and are compact operators. (As and , it is sufficient to require that only one of and be compact.) We denote the space of all almost periodic functionals in by .
An operator space that is also an algebra such that multiplication is completely contractive, is called a completely contractive Banach algebra.
We define:
Definition 2.1.
Let be a completely contractive Banach algebra. We call completely almost periodic if and denote the collection of completely almost periodic functionals in by .
Remarks.

Obviously, is a closed linear subspace of .

Since Schauder’s theorem is no longer true for complete compactness, we do need the requirement that both and are completely compact.

Since and , we could have replaced in complete almost periodicity—based on complete compactness—in Definition 2.1 by the analogous notion involving Gelfand complete compactness instead: we would still have obtained the same functionals.
We shall now look at a special class of completely contractive Banach algebras which arise naturally in abstract harmonic analysis.
Recall that a Hopf–von Neumann algebra is a pair where is a von Neumann algebra and is a comultiplication, i.e., a normal, unital, homomorphism satisfying
The comultiplication induces a product on via
which turns into a completely contractive Banach algebra.
Examples.

Let be a locally compact group, and define a comultiplication —noting that —by letting
Then with the product induced by is just the usual convolution algebra . Since the operator space structure on is maximal, equals ; it consists precisely of the almost periodic continuous functions on in the classical sense (see [Bur], for instance) and thus, in particular, is a subalgebra of .

Let again be a locally compact group, let be its group von Neumann algebra, i.e., the von Neumann algebra generated by , where is the left regular representation of on , and let
Then is Eymard’s Fourier algebra ([Eym]), and the product induced by is pointwise multiplication. The space —often denoted by —was first considered in [D–R] and further studied in [Gra 1] and [Lau], for instance.
Except in a few special cases, e.g., if is abelian or discrete and amenable (as a consequence of [Gra 1, Theorem 12] and [Gra 2, Proposition 2]), is unknown whether or not is a subalgebra of .
The picture changes once we replace almost periodicity with complete almost periodicity:
Theorem 2.2.
Let be a Hopf–von Neumann algebra such that is injective. Then is a subalgebra of .
Proof.
Let . Then, by Proposition 1.4, is completely compact if and only if it is a norm limit of finite rank operators. In view of the completely isometric identifications ([E–R, Corollary 7.1.5] and [E–R, Theorem 7.2.4])
and of [E–R, Proposition 8.1.2], this means that is completely compact if and only if . A similar assertion holds for .
All in all, we have that
Since is nothing but the spatial tensor product of with itselt and thus subalgebra of , and since is a homomorphism, this proves the claim. ∎
If is an amenable or connected locally compact group, then is well known to be injective. Writing for , we thus have:
Corollary 2.3.
Let be an amenable or connected locally compact group. Then is a subalgebra of .
Remarks.

If is abelian—or, more generally, finitebyabelian—, then the canonical operator space structure on is equivalent to , so that .
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[March 3, 2021] Author’s address: Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta Canada T6G 2G1 Email: URL: http://www.math.ualberta.ca/runde/