Polynomial growth and functional calculus in algebras of integrable cross-sections

A great deal of effort within the subject of abstract harmonic analysis has been put into understanding the $L^{1}$-algebras and weighted $L^{1}$-algebras of locally compact groups. Most of this research has focused into extending properties of $L^{1}(\mathbb{R})$ or $\ell^{1}(\mathbb{Z})$ to more general algebras. In general, one could say that the case of abelian groups is fairly well understood [43] and teaches us what to expect for more general algebras, like $L^{1}({\sf G})$, for a noncommutative group ${\sf G}$.

In particular, we are going to be interested in spectral properties like symmetry [29, 34], norm-controlled inversion [21, 22] and ideal-theoretic properties like ^∗-regularity [8, 10, 33] or the Wiener property [23, 28, 34]. This properties will be defined at the right time. For now, let us vaguely state that symmetry means that the spectral theory of our algebra is that of a $C^{*}$-algebra, with ^∗-regularity being an analogous statement about the primitive space. The property of norm-controlled inversion holds if the norms of inverses can be estimated using $C^{*}$-norms and the Wiener property is a generalization of Wiener’s tauberian theorem [47]. Our reference for the theory of Banach ^∗-algebras is [39].

The case of groups with polynomial growth is already interesting enough, as some of the properties previously mentioned no longer hold in full generality (like symmetry). In this setting, the ideal theory of group algebras has been studied on [5, 35] (mostly under the assumption of symmetry) and the case of weighted group algebras on [15, 19, 36]. Norm-controlled inversion was studied for group algebras in [38, 44]. Let us also mention that a ^∗-regular algebra has a unique $C^{*}$-norm and the question of uniqueness of $C^{*}$-norms on group algebras (or even group rings) has been studied in [12, 1].

It seems natural to us that a continuation of these studies would consider group algebras twisted by a cocycle $L_{\omega}^{1}({\sf G})$, convolution algebras $L_{\alpha}^{1}({\sf G},\mathfrak{A})$ associated with a $C^{*}$-dynamical system $({\sf G},\mathfrak{A},\alpha)$, or more generally the algebra of cross-sections $L^{1}({\sf G}\,|\,\mathscr{C})$ associated with a Fell bundle $\mathscr{C}\overset{q}{\to}{\sf G}$. However, the current literature is remarkably lacking in this respect. A lot of progress has been made in the study of symmetry [20, 26, 27, 29], but results about the Wiener property or ^∗-regularity of these algebras are few and apart [30, 31] (we remark that most of these results are based on the construction of a functional calculus beyond the one based on holomorphic functions -naturally existing in any Banach ^∗-algebra-). In particular, anything stated generally enough and beyond the algebras of finite dimensional $C^{*}$-dynamical systems seems to be new. Let us now exemplify some of the difficulty that this problem posed.

Now, in any more general case involving algebras with coefficients other than complex numbers, the naive approach to recreating these arguments completely breaks down. If followed without any changes, one would obtain estimates involving the norm in $L^{2}({\sf G}\,|\,\mathscr{C})$, which is not a Hilbert $C^{*}$-module, so there is no hope of using $C^{*}$-algebra theory. On the other hand, if we replace $L^{2}({\sf G}\,|\,\mathscr{C})$ by the usually chosen Hilbert $C^{*}$-module -here denoted as $L^{2}_{\sf e}({\sf G}\,|\,\mathscr{C})$-, no progress can be made after the first inequalities. This exemplifies some lack of interplay between the $L^{p}$ norms and the norm of $L^{2}_{e}({\sf G}\,|\,\mathscr{C})$ and poses a major problem when trying to pass from group algebras to algebras of a crossed-product-type. In simpler terms, to progress, one would like to majorize an $L^{p}$-norm with a quantity involving the $L^{2}_{e}$-norm. The $L^{2}_{e}$-norm, however, is smaller than the $L^{2}$-norm.

This motivated the authors in [32] to define new norms with a more harmonious interplay between them (their $2$-norm being the same norm of $L^{2}_{\sf e}({\sf G}\,|\,\mathscr{C})$) and essentially abandon the study of the classical $L^{1}$-algebras. We take their norms and note they could be used to prove an strengthened version of Young’s inequality, at least in the case $r=\infty,p=q=2$. This fact, together with others (Lemma 3.4) is then used to bypass the technical difficulties and relate the Hilbert $C^{*}$-module norm to the $L^{p}$-norms just enough to get results. This lack of access to inequalities together with the natural difficulties associated to our level of generality (which include the fact that the functions of $L^{1}({\sf G}\,|\,\mathscr{C})$ do not take values in a single Banach space) make our proofs rather indirect and different from the classical ones.

In any case, we are able to produce far-reaching generalizations that we compile in the following theorem.

We remark that Theorem 1.3 is already new when $\mathfrak{D}=L_{\alpha}^{1}({\sf G},\mathfrak{A})$ and $({\sf G},\mathfrak{A},\alpha)$ is a $C^{*}$-dynamical system and thus our results greatly extend the previous literature. Let us also mention that $L^{1}({\sf G}\,|\,\mathscr{C})$ is symmetric when ${\sf G}$ is nilpotent [20] or compact (Theorem 7.5), while $L^{1,\nu}({\sf G}\,|\,\mathscr{C})$ is always symmetric (Theorem 4.14). However, we only guarantee the existence of such a weight in the case that ${\sf G}$ is compactly generated (Proposition 4.6) or when ${\sf G}$ can be saturated by an increasing sequence of compact subgroups (Example 4.7).

Based on the inequalities we proved, we can also produce the following result about norm-controlled inversion. It seems to be the first result known for convolution algebras where the algebra of coefficients is not $\mathbb{C}$.

We now briefly describe the content of the article. Section 2 contains preliminaries. We mostly fix notation and prove the Dixmier-Baillet theorem, which we use to construct our functional calculus. Section 3 deals with proving Lemma 3.4 (our main computational lemma) and deriving the growth estimate of the semigroups $e^{it\Phi}$ in the case of $L^{1}({\sf G}\,|\,\mathscr{C})$. In Section 4 we show the same but for $L^{1,\nu}({\sf G}\,|\,\mathscr{C})$, after showing that this algebra is both symmetric and inverse-closed in $L^{1}({\sf G}\,|\,\mathscr{C})$. In Section 5 we derive the rest of Theorem 1.3, based on the results we obtained and some well-established methods. Each part of theorem is treated in a different subsection. In these subsections we also derive some other (smaller) properties of these algebras, not mentioned here. Subsection 5.4 is somewhat different to the rest, as it does not rely on the functional calculus we developed, but on the norm estimates. It is devoted to introduce the algebra $\mathfrak{E}$ and prove Theorem 1.4. Section 6 is dedicated to show that twisted Hahn algebras of transformation groupoids where the acting group has polynomial growth are ^∗-regular, upgrading the existing results (of $C^{*}$-uniqueness) in this case. Finally, Section 7 is an appendix, where we use Lemma 3.4 to show that $C_{\rm c}({\sf G}\,|\,\mathscr{C})$ is quasi-symmetric in the subexponential growth case and that compact groups are hypersymmetric. This last fact both extends the applicability of our main theorem and helps us provide the first known examples of hypersymmetric groups with non-symmetric discretizations.

Let $\mathfrak{B}$ be a Banach algebra. $\mathfrak{B}(b_{1},\ldots,b_{n})$ denotes the closed subalgebra of $\mathfrak{B}$ generated by the elements $b_{1},\ldots,b_{n}\in\mathfrak{B}$. The set of invertible elements in $\mathfrak{B}$ is denoted by ${\rm Inv}(\mathfrak{B})$. If $\mathfrak{B}$ has an involution, $\mathfrak{B}_{\rm sa}$ denotes the set of self-adjoint elements in $\mathfrak{B}$. The Banach algebra of bounded operators over the Banach space $\mathcal{X}$ is denoted by $\mathbb{B}(\mathcal{X})$. If $\mathfrak{B}$ is a commutative Banach algebra with spectrum $\Delta$, then $\hat{b}\in C_{0}(\Delta)$ denotes the Gelfand transform of $b\in\mathfrak{B}$. In particular, the Fourier transform of a complex function $f:\mathbb{R}\to\mathbb{C}$ is

As usual, ${\rm Spec}_{\mathfrak{B}}(b)=\{\lambda\in\mathbb{C}\mid b-\lambda 1\textup{ is not invertible in }\widetilde{\mathfrak{B}}\}$ will denote the spectrum of an element $b\in\mathfrak{B}$, while

denotes its spectral radius. Gelfand’s formula for the spectral radius says that $\rho_{\mathfrak{B}}(b)=\lim_{n\to\infty}\lVert b^{n}\rVert_{\mathfrak{B}}^{1/n}$.

The key idea for us is that the self-adjoint elements of polynomial growth admit a smooth functional calculus. The concrete fact is the following theorem, attributed to Dixmier [14, Lemme 7] and Baillet [4, Théorème 1]. See also [32, Theorem 1.3]. Let us also mention that functional calculi involving smooth functions has also appeared in other (different, but related) contexts, such as [25, 37].

From now on ${\sf G}$ will be a (Hausdorff) unimodular, locally compact group with unit ${\sf e}$ and Haar measure $d\mu(x)\equiv dx$. If ${\sf G}$ is compact, we assume that $\mu$ is normalized so that $\mu({\sf G})=1$. During most of the article, ${\sf G}$ will be assumed of polynomial growth of order $d$, meaning that

for all relatively compact subsets $K\subset{\sf G}$. We will also fix a Fell bundle $\mathscr{C}\!=\bigsqcup_{x\in{\sf G}}\mathfrak{C}_{x}$ over ${\sf G}$. The algebra of integrable cross-sections $L^{1}({\sf G}\,|\,\mathscr{C})$ is a Banach ^∗-algebra and a completion of the space $C_{\rm c}({\sf G}\,|\,\mathscr{C})$ of continuous sections with compact support. Its (universal) $C^{*}$-algebra its denoted by ${\rm C^{*}}({\sf G}\,|\,\mathscr{C})$. For the general theory of Fell bundles we followed [18, Chapter VIII], to which we refer for details. We will only recall the product on $L^{1}({\sf G}\,|\,\mathscr{C})$, given by

and its involution

in terms of the operations $\big{(}\bullet,^{\bullet}\big{)}$ on the Fell bundle. We will also consider the $L^{p}$-spaces $L^{p}({\sf G}\,|\,\mathscr{C})$, endowed with the norms

If ${\sf G}$ is compact, $L^{p}({\sf G}\,|\,\mathscr{C})$ is a dense Banach ^∗-subalgebra of $L^{1}({\sf G}\,|\,\mathscr{C})$. In particular $C({\sf G}\,|\,\mathscr{C})$ is also a Banach ^∗-algebra when endowed with the $L^{\infty}$-norm. Is immediate that $p\geq q$ implies $L^{q}({\sf G}\,|\,\mathscr{C})\subset L^{p}({\sf G}\,|\,\mathscr{C})$. On the other hand, if ${\sf G}$ is discrete, we will write $\ell^{p}({\sf G}\,|\,\mathscr{C})$ instead of $L^{p}({\sf G}\,|\,\mathscr{C})$.

We now turn our attention to the representations and $C^{*}$-algebras. In order to define a reduced $C^{*}$-completion, we would like to use the representation of $L^{1}({\sf G}\,|\,\mathscr{C})$ as bounded operators on $L^{2}({\sf G}\,|\,\mathscr{C})$ and this would allow us to use the usual interplay between the $L^{p}$-norms (Hölder’s, Young’s inequality, etc) to deduce facts about $L^{1}({\sf G}\,|\,\mathscr{C})$. However, $L^{2}({\sf G}\,|\,\mathscr{C})$ is not a Hilbert $C^{*}$-module, so it seems unlikely that we can get a $C^{*}$-algebra. This forces us to consider the space $L^{2}_{\sf e}({\sf G}\,|\,\mathscr{C})$, the completion of $L^{2}({\sf G}\,|\,\mathscr{C})$ under the norm

This is a Hilbert $C^{*}$-module over $\mathfrak{C}_{\sf e}$, so the set of adjointable operators is a $C^{*}$-algebra under the operator norm. We denote this algebra by $\mathbb{B}_{a}(L^{2}_{\sf e}({\sf G}\,|\,\mathscr{C}))$. The left regular representation $\lambda$ is then the ^∗-monomorphism given by

It follows from the results in [17] that, for amenable groups, the universal $C^{*}$-completion (the so-called $C^{*}$-envelope) of $L^{1}({\sf G}\,|\,\mathscr{C})$, denoted ${\rm C^{*}}({\sf G}\,|\,\mathscr{C})$, coincides with $\overline{\lambda(L^{1}({\sf G}\,|\,\mathscr{C}))}^{\lVert\cdot\rVert_{\mathbb{B}_{a}(L^{2}_{\sf e}({\sf G}\,|\,\mathscr{C}))}}$. Now we would like to do some spectral analysis, but in order to do so, we feel the need to define $L^{p}$ versions of the norm $\lVert\cdot\rVert_{L^{2}_{\sf e}({\sf G}\,|\,\mathscr{C})}$. Our inspiration then comes from [32, Definition 2.1].

The next lemma compiles some facts about the norms $\lVert\cdot\rVert_{\pi,p}$ and their relations with the other norms previously introduced. It can be regarded as both our new approach to the problem and our main computation tool.