Extending representations of Banach algebras to their biduals

TL;DR

This paper characterizes when representations of Banach algebras extend to their biduals, linking weak compactness of orbit maps to the structure of representations and their applications in C*-algebras and harmonic analysis.

Contribution

It provides necessary and sufficient conditions for extending Banach algebra representations to their biduals, and explores implications for C*-algebras and $L^p$-space representations.

Findings

Representation extension characterized by weakly compact orbit maps.

Essential space of representation is complemented under certain conditions.

C*-algebra $A$ has an isometric $L^p$-space representation iff $A$ is commutative.

Abstract

We show that a representation of a Banach algebra $A$ on a Banach space $X$ can be extended to a canonical representation of $A^{**}$ on $X$ if and only if certain orbit maps $A\to X$ are weakly compact. When this is the case, we show that the essential space of the representation is complemented if $A$ has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in C*-algebras and in abstract harmonic analysis. For example, a C*-algebra $A$ has an isometric representation on an $L^p$-space, for $p\in[1,\infty)\setminus\{2\}$, if and only if $A$ is commutative. Moreover, the $L^p$-operator algebra of a locally compact group is universal with respect to arbitrary representations on $L^p$-spaces.

Equations

$$A^{\ast \ast }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(A^{\ast \ast },◊)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X^{\ast \ast }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{\phi }(S)(x)={\mathrm{ev}}_x^{\ast \ast }({\phi }^{\ast \ast }(S))=({\mathrm{ev}}_x\circ \phi {)}^{\ast \ast }(S),$$

$$\tilde{\phi }(S)x=({\mathrm{ev}}_x\circ \phi {)}^{\ast \ast }(S)\in X_{\phi },$$

$$\tilde{\phi }(a)\tilde{\phi }(b)={\alpha }_X({\phi }^{\ast \ast }(a)){\alpha }_X({\phi }^{\ast \ast }(b))={\phi }^{\ast \ast }(a){\alpha }_X({\phi }^{\ast \ast }(b))={\alpha }_X({\phi }^{\ast \ast }(a\square b)),$$

$$\Vert p\Vert =\Vert \tilde{\phi }(e)\Vert \le \Vert \tilde{\phi }\Vert \Vert e\Vert =\Vert \phi \Vert \Vert e\Vert .$$

$$p\tilde{\phi }(S)=\tilde{\phi }(e)\tilde{\phi }(S)=\tilde{\phi }(e◊S)=\tilde{\phi }(S).\text{qed}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{\phi }(T)(\phi (a)(x))=\phi (T(a))(x).$$

$$\phi (T(a))(x)=\underset{j\in J}{\lim }\phi (T(e_{j}a))(x)=\underset{j\in J}{\lim }\phi (T(e_j)a)(x)=\underset{j\in J}{\lim }\phi (T(e_j))(\phi (a)(x)).$$

$$\phi (T(b))(y)=\underset{j\in J}{\lim }\phi (T(e_j))(\phi (b)(y)).$$

$$\tilde{\phi }(ST)(\phi (a)(x))=\phi (ST(\phi (a)))(x)=\tilde{\phi }(S)(\phi (T(\phi (a)))(x))=\tilde{\phi }(S)\tilde{\phi }(T)(\phi (a)(x)),$$

$$\tilde{\phi }(L_a)(\phi (b)(x))=\phi (L_a(b))(x)=\phi (ab)(x)=\phi (a)(\phi (b)(x)),$$

$$\Vert \tilde{\phi }(T)\Vert$$

$$\displaystyle=\sup\big{\{}\|\widetilde{\varphi}(T)(\varphi(a)(x))\|:a\in A,x\in X,\|a\|\leq 1,\|x\|<1\big{\}}$$

$$\displaystyle=\sup\big{\{}\|\varphi(T(a))(x)\|:a\in A,x\in X,\|a\|\leq 1,\|x\|<1\big{\}}$$

$$\le \Vert T\Vert ,$$

$$\Vert T\Vert$$

$$\displaystyle=\sup\big{\{}\|\varphi(T(a)(x))\|:a\in A,x\in X,\|a\|\leq 1,\|x\|\leq 1\big{\}}$$

$$\displaystyle=\sup\big{\{}\|\widetilde{\varphi}(T)(\varphi(a)x)\|:a\in A,x\in X,\|a\|\leq 1,\|x\|\leq 1\big{\}}$$

$$\le \Vert \tilde{\phi }(T)\Vert .\text{qed}$$

$$\Vert \exp (it\phi (a))\Vert =\Vert \phi (\exp (ita))\Vert \le \Vert \exp (ita)\Vert =1,$$

$$(v\xi )(x)=h_v(x){\left(\frac{d(\mu \circ T_v^{-1})}{d\mu }(x)\right)}^{\frac{1}{p}}\xi (T_v^{-1}(x))$$

$$\|a\|_{\mathcal{E}}=\sup\big{\{}\|\varphi(a)\|:\varphi\in\operatorname{Rep}_{\mathcal{E}}(A)\big{\}}.$$

$$\|a\|_{\mathcal{E}}=\sup\big{\{}\|\varphi(a)\|:\varphi\colon A\to{\mathcal{L}}(X)\text{ contractive representation},X\in\mathcal{E}\big{\}}.$$

$$\Vert \phi (a)\Vert =\Vert {\phi }_0(a)\Vert \le \Vert a{\Vert }_{\mathcal{E}}.\text{qed}$$

Full text

Our results have interesting consequences both in $C^{*}$-algebras and in abstract harmonic analysis. For example, a $C^{*}$-algebra $A$ has an isometric representation on an $L^{p}$-space, for $p\in[1,\infty)\setminus\{2\}$, if and only if $A$ is commutative. Moreover, the $L^{p}$-operator algebra of a locally compact group is universal with respect to arbitrary representations on $L^{p}$-spaces.

Part of this research was conducted while the authors were taking part in the Research Program Classification of operator algebras, complexity, rigidity and dynamics, held at the Institut Mittag-Leffler, between January and April of 2016. The first named author was partially supported by a Postdoctoral Research Fellowship from the Humboldt Foundation. The authors were partially supported by the Deutsche Forschungsgemeinschaft (SFB 878 Groups, Geometry & Actions).

1. Introduction

Let $A$ be a Banach algebra. By the groundbreaking work of Arens, multiplication on $A$ can be extended in two natural ways to a multiplication on its bidual $A^{**}$, called the left Arens product $\square$ and the right Arens product $\lozenge$. Each of these multiplications give $A^{**}$ the structure of a Banach algebra, and the map $\kappa_{A}\colon A\to A^{\ast\ast}$ is multiplicative. Despite being complicated objects, biduals of Banach algebras are usually accessible through powerful tools that are not at one’s disposal to study the Banach algebras themselves: among others, arguments involving polar decompositions can be performed, and a much more general type of functional calculus is available. The technical advantages enjoyed by biduals are regularly exploited in the theory of $C^{*}$-algebras, where Kaplansky’s density theorem often allows one to transfer information back to $A$.

It therefore becomes relevant to know in what situations one can work in the much more flexible context of the bidual algebra. More concretely, we are interested in knowing when a representation of $A$ extends to $A^{\ast\ast}$.

Problem \thepbmCt.

Given a representation $\varphi\colon A\to{\mathcal{L}}(X)$, find conditions that guarantee that there exists a representation $\tilde{\varphi}\colon(A^{**},\lozenge)\to{\mathcal{L}}(X)$, or $\tilde{\varphi}\colon(A^{**},\square)\to{\mathcal{L}}(X)$, making the following diagram commute:

[TABLE]

As stated, this problem is very general and a complete answer seems out of reach. However, in Section 2 we obtain a complete answer if we require the extension to be the ‘canonical’ one, in the following sense. In [GT16a], we introduced a natural multiplication on ${\mathcal{L}}(X,X^{**})$ and a multiplicative operator $\alpha_{X}\colon({\mathcal{L}}(X)^{**},\lozenge)\to{\mathcal{L}}(X,X^{**})$; see Section 2 for details.

The composition $\alpha_{X}\circ\varphi^{**}\colon A^{**}\to{\mathcal{L}}(X,X^{**})$ is a canonical candidate for a solution to Section 1; the only condition that may fail is that its range is not always contained in ${\mathcal{L}}(X)$. Besides being natural, this map has the remarkable feature that, whenever it is a solution to the extension problem, then it is in fact multiplicative with respect to both Arens products. Other particular aspects of this canonical solution are exploited in Section 3; see also Theorem B below.

In the main result of this work, we characterize when a canonical extension exists:

Theorem A**.**

Let $\varphi\colon A\to{\mathcal{L}}(X)$ be a representation. Then Section 1 has a canonical solution (for either Arens product, or both) if and only if the orbit map $\mathrm{ev}_{x}\circ\varphi\colon A\to X$ is weakly compact for all $x\in X$.

The easiest situation to apply Section 2 is when every operator $A\to X$ is weakly compact. In this case, every representation of $A$ on $X$ has a canonical extension to a representation of $A^{**}$ (endowed with either Arens product) on $X$. There are many interesting situations in which every operator $A\to X$ is weakly compact: If $A$ or $X$ is reflexive; or if $A$ is a $C^{*}$-algebra and $X$ does not contain an isomorphic copy of $c_{0}$; see Section 2.

If $\varphi\colon A\to{\mathcal{L}}(X)$ is a representation, we define its essential space $X_{\varphi}$ as the closed linear span of $\{\varphi(a)x\colon a\in A,x\in X\}$, and we say that $\varphi$ is nondegenerate if $X_{\varphi}=X$. Nondegeneracy is a technically important condition, which grants one access to extension arguments involving multiplier algebras. For a locally compact group $G$, nondegeneracy is precisely the assumption needed to be able to ‘disintegrate’ a representation $L^{1}(G)\to{\mathcal{L}}(X)$ to an action of $G$ on $X$.

In practice, however, one often encounters nondegenerate representation. The difference is relatively insignificant for representations on Hilbert spaces, since one can simply restrict the attention to the essential space (which is again a Hilbert space). For a more general Banach space $X$, the distinction is more important. It is particularly valuable to know when $X_{\varphi}$ is complemented in $X$, since many properties of Banach spaces pass to complemented subspaces. In Section 3, we use Theorem A to show that this is often the case:

Theorem B**.**

Let $A$ be a Banach algebra with a bounded left approximate indentity, and let $\varphi\colon A\to{\mathcal{L}}(X)$ be a representation. If the orbit map $\mathrm{ev}_{x}\circ\varphi\colon A\to X$ is weakly compact for all $x\in X$, then $X_{\varphi}$ is complemented in $X$.

We make a few comments about the proof of Theorem B. When $A$ has a bounded left approximate identity, the Banach algebra $(A^{**},\lozenge)$ has a left unit $e$; see Section 3. Thus, if the representation $\varphi\colon A\to{\mathcal{L}}(X)$ has an extension to some representation $\tilde{\varphi}\colon(A^{**},\lozenge)\to{\mathcal{L}}(X)$, then $\tilde{\varphi}(e)$ is a natural candidate for a projection onto the essential space. In general, the range of the projection $\tilde{\varphi}(e)$ can be strictly bigger than the essential space of $\varphi$. Nevertheless, and somewhat surprisingly, we show that this is not the case for the canonical extensions from Theorem A. We also obtain bounds on the norm of the projection, which allows us to show that the essential space is $1$-complemented if $\varphi$ is contractive and $A$ has a contractive left approximate identity; see Section 3.

In Section 5 and Section 6 we present two applications of our results to what are arguably the most relevant classes of Banach algebras: $C^{*}$-algebras and group algebras. Concretely, we show in Section 5 that a $C^{*}$-algebra can be isometrically represented on some $L^{p}$-space, for $p\in[1,\infty)\setminus\{2\}$, if and only if it is commutative. This result is to some extent unexpected, and it should be compared with the fact that every $C^{*}$-algebra can be represented on a noncommutative $L^{p}$-space, for any $p\in[1,\infty)$; see Section 5.

Let $G$ be a locally compact group. The completion of $L^{1}(G)$ for nondegenerate representations on $L^{p}$-spaces is called the universal group $L^{p}$-operator algebra, denoted $F^{p}(G)$; see [Phi13, GT15b]. The Banach algebra $F^{p}(G)$ captures the isometric representation theory of $G$ on $L^{p}$-spaces, and is universal with respect to nondegenerate representations of $L^{1}(G)$ on $L^{p}$-spaces. Using Theorem B, we show that $F^{p}(G)$ is universal for all representations of $L^{1}(G)$ on $L^{p}$-spaces; see Section 6. The results of this paper, in particular Section 5 and Section 6, have been applied in [GT16c] to give a complete answer to the question of when $F^{p}(G)$ and $F^{q}(G)$ are isometrically isomorphic, among others.

Acknowledgements

The authors would like to thank Philip G. Spain for valuable electronic correspondence. The authors would like to thank the staff and organizers, and Søren Eilers in particular, for the hospitality during their visits to the Research Program Classification of operator algebras, complexity, rigidity and dynamics, held at the Institut Mittag-Leffler, between January and April of 2016.

2. Extending representations of a Banach algebra to its bidual

Throughout this section, $A$ denotes a Banach algebra, $X$ denotes a Banach space, and $\varphi\colon A\to{\mathcal{L}}(X)$ denotes a representation. Using the multiplicative operator $\alpha_{X}\colon({\mathcal{L}}(X)^{**},\lozenge)\to{\mathcal{L}}(X,X^{**})$ constructed in [GT16a], we extend $\varphi$ to a multiplicative operator $\tilde{\varphi}\colon(A^{**},\lozenge)\to{\mathcal{L}}(X,X^{**})$; see Section 2 and Section 2. The main result of this section is Section 2, where we characterize when the image of $\tilde{\varphi}$ is contained in ${\mathcal{L}}(X)$ in terms of weak compactness of orbit maps $A\to X$. If this is the case, it follows that $\widetilde{\varphi}$ is multiplicative not only with respect to $\lozenge$, but also with respect to $\square$.

\thepgrCt**.**

Given operators $a,b\colon X\to X^{**}$, their product $ab$ is defined as the composition $\kappa_{X^{*}}^{*}\circ a^{**}\circ b$. This gives ${\mathcal{L}}(X,X^{**})$ the structure of a unital Banach algebra, with unit $\kappa_{X}$; see [GT16a, Paragraph 3.9]. Note that ${\mathcal{L}}(X,X^{**})$ is anti-isomorphic to ${\mathcal{L}}(X^{*})$ by [GT16a, Proposition 3.11]

We let $\gamma_{X}\colon{\mathcal{L}}(X)\to{\mathcal{L}}(X,X^{**})$ be given by $\gamma_{X}(a)=\kappa_{X}\circ a$ for $a\in{\mathcal{L}}(X)$. Then $\gamma_{X}$ is an isometric, multiplicative operator; see [GT16a, Paragraph 3.11].

Let $\alpha_{X}\colon{\mathcal{L}}(X)^{**}\to{\mathcal{L}}(X,X^{**})$ be the contractive operator introduced in [GT16a, Definition 3.16]. Instead of recalling the definition of $\alpha_{X}$, we provide a formula to compute $\alpha_{X}(S)$ for $S\in{\mathcal{L}}(X)^{**}$; see Lemma 2.2. (For the purposes of this paper, Lemma 2.2 can be taken as the definition of $\alpha_{X}$.)

In general, the bitranspose of a multiplicative operator between Banach algebras is again multiplicative if both bidual Banach algebras are equipped with the same Arens product. Thus, $\varphi^{**}\colon(A^{**},\lozenge)\to({\mathcal{L}}(X)^{**},\lozenge)$ is multiplicative. The map $\alpha_{X}$ is multiplicative for the right Arens product $\lozenge$ on ${\mathcal{L}}(X)^{**}$, and we have $\gamma_{X}=\alpha_{X}\circ\kappa_{{\mathcal{L}}(X)}$; see Corollary 3.23 and Lemma 3.19 in [GT16a]. We thus obtain the following commutative diagram, where each map is a multiplicative operator:

[TABLE]

Proposition \theprpCt.

The map $\tilde{\varphi}=\alpha_{X}\circ\varphi^{**}\colon A^{**}\to{\mathcal{L}}(X,X^{**})$ is multiplicative for the right Arens product $\lozenge$ on $A^{**}$. Moreover, we have $\|\tilde{\varphi}\|=\|\varphi\|$.

Proof.

It remains to verify that $\|\tilde{\varphi}\|=\|\varphi\|$. By [GT16a, Remark 3.17], the map $\alpha_{X}$ is contractive, so $\|\tilde{\varphi}\|\leq\|\varphi\|$. The converse estimate follows using that $\gamma_{X}\circ\varphi=\tilde{\varphi}\circ\kappa_{A}$ and that $\gamma_{X}$ is isometric. ∎

Recall that an operator $f\colon X\to Y$ is said to be weakly compact if the image of the unit ball of $X$ is contained in a weakly compact subset of $Y$. By Gantmacher’s theorem, $f$ is weakly compact if and only if its transpose $f^{*}$ is. Moreover, $f$ is weakly compact if and only if the image of $f^{**}\colon X^{**}\to Y^{**}$ is contained in $Y$; see [Con90, Theorem VI.5.5, p.185].

Lemma 2.1**.**

Let $X$ and $Y$ be Banach spaces, let $Y_{0}\subseteq Y$ be a closed subspace, and let $f\colon X\to Y$ be an operator whose image is contained in $Y_{0}$. Then $f$ is weakly compact if and only if the image of $f^{**}\colon X^{**}\to Y^{**}$ is contained in $Y_{0}$.

Proof.

If the image of $f^{**}$ is contained in $Y_{0}$, then it is also contained in $Y$, and consequently $f$ is weakly compact. To show the backward implication, assume that $f$ is weakly compact. Let $\iota\colon Y_{0}\to Y$ be the inclusion map. Using that the image of $f$ is contained in $Y_{0}$, we let $f_{0}\colon X\to Y_{0}$ be the unique map such that $f=\iota\circ f_{0}$.

We identify $Y_{0}^{**}$ with a subspace of $Y^{**}$, and $Y$ with a subspace of $Y^{**}$. There is a commutative diagram:

[TABLE]

We claim that $Y_{0}=Y_{0}^{**}\cap Y$ in $Y^{**}$. The inclusion ‘$\subseteq$’ is clear. To show the reverse inclusion, let $y\in Y_{0}^{**}\cap Y$. Choose a net $(y_{j})_{j\in J}$ in $Y_{0}$ that converges weak∗ to $y$ in $Y_{0}^{**}$. Since the inclusion $\iota^{**}$ is weak∗ continuous, we deduce that $(y_{j})_{j\in J}$ converges weak∗ to $y$ in $Y^{**}$. Since the net $(y_{j})_{j\in J}$ and $y$ belong to $Y$, it follows that $(y_{j})_{j\in J}$ converges weakly to $y$. It follows from the Hahn-Banach theorem that (norm-)closed subspaces are weakly closed, which implies that $y$ belongs to $Y_{0}$, which proves the claim.

Since $f$ is weakly compact, the image of $f^{**}$ is contained in $Y$. Using that $\iota^{**}$ is isometric, it follows that the image of $f^{**}$ is contained in the image of $f_{0}^{**}$, which in turn is contained in $Y_{0}^{**}$. Using the claim, the result follows. ∎

The following is the case $X=Y$ of [GT16a, Lemma 3.18].

Lemma 2.2**.**

For $x\in X$, let ${\mathrm{ev}}_{x}\colon{\mathcal{L}}(X)\to X$ be the evaluation map. Then $\alpha_{X}(S)(x)={\mathrm{ev}}_{x}^{**}(S)$ for all $S\in{\mathcal{L}}(X)^{**}$ and $x\in X$.

We consider ${\mathcal{L}}(X)$ as a subalgebra of ${\mathcal{L}}(X,X^{**})$ via the map $\gamma_{X}$. The following consequence is our desired (partial) answer to Section 1.

Theorem \thethmCt.

Let $\varphi\colon A\to{\mathcal{L}}(X)$ be a representation of a Banach algebra $A$ on a Banach space $X$, and set $\tilde{\varphi}=\alpha_{X}\circ\varphi^{**}\colon A^{**}\to{\mathcal{L}}(X,X^{**})$. Then the image of $\tilde{\varphi}$ is contained in ${\mathcal{L}}(X)$ if and only if for every $x\in X$, the orbit map ${\mathrm{ev}}_{x}\circ\varphi\colon A\to X$ is weakly compact. Moreover, if this is the case, then:

1. (1)

The essential spaces of $\varphi$ and $\tilde{\varphi}$ agree. 2. (2)

$\widetilde{\varphi}\colon A^{**}\to{\mathcal{L}}(X)$* is multiplicative with respect to $\lozenge$ and $\square$.* 3. (3)

$\widetilde{\varphi}$* is an extension of $\varphi$.*

Proof.

Using Lemma 2.2 at the first step, we have

[TABLE]

for all $S\in A^{**}$ and $x\in X$. Thus, given $x\in X$, the orbit map ${\mathrm{ev}}_{x}\circ\varphi$ is weakly compact if and only if $\tilde{\varphi}(S)(x)$ belongs to ${\mathcal{L}}(X)$ for every $S\in A^{**}$, by the comments before Lemma 2.1. This implies the first statement.

From now on, we assume that every orbit map ${\mathrm{ev}}_{x}\circ\varphi$ is weakly compact, and hence that the image of $\widetilde{\varphi}$ is contained in the image of $\gamma_{X}$.

(1). We clearly have $X_{\varphi}\subseteq X_{\tilde{\varphi}}$. Conversely, given $x\in X$, since ${\mathrm{ev}}_{x}\circ\varphi$ is weakly compact with image contained in $X_{\varphi}$, it follows from Lemma 2.1 that the image of $({\mathrm{ev}}_{x}\circ\varphi)^{**}$ is contained in $X_{\varphi}$ as well. Using Lemma 2.2 at the first step, we deduce that

[TABLE]

for every $S\in A^{**}$ and $x\in X$. Hence, $X_{\tilde{\varphi}}\subseteq X_{\varphi}$, as desired.

(2). That $\widetilde{\varphi}$ is multiplicative with respect to $\lozenge$ follows from Section 2. Given $a,b\in A$, we use the second identity of Lemma 3.20 of [GT16a] at the second step (with $T=\varphi^{**}(a)$ and $S=\widetilde{\varphi}(b)\in\gamma_{X}({\mathcal{L}}(X))$), and Lemma 3.21 of [GT16a] at the third step, to get

[TABLE]

as desired.

(3). This is immediate, since $\alpha_{X}\circ\kappa_{{\mathcal{L}}(X)}=\gamma_{X}$. ∎

\thepgrCt**.**

If $X$ and $Y$ are Banach spaces, there are some very general situations in which every operator $X\to Y$ is automatically weakly compact. For example, if either $X$ or $Y$ is reflexive (since a Banach space $E$ is reflexive if and only if ${\mathrm{id}}_{E}$ is weakly compact). However, there are situations where neither $X$ nor $Y$ is reflexive, yet every operator $X\to Y$ is weakly compact. This is for instance the case when $X$ has Pełczyński’s property $(V)$ and $Y$ contains no isomorphic copy of $c_{0}$; this is essentially an immediate consequence of [BP58, Theorem 5].

By a deep result of Pfitzner, [Pfi94, Carollary 6], every $C^{*}$-algebra $A$ has property $(V)$. It follows that if $Y$ contains no isomorphic copy of $c_{0}$ then every operator $A\to Y$ is weakly compact. This had previously been obtained in [ADG72, Theorem 4.2].

The class of spaces that do not contain an isomorphic copy of $c_{0}$ is known to include all reflexive space, and all $L^{p}$-spaces for $p\in[1,\infty)$.

Since $C^{*}$-algebras are Arens regular, we do not have to specify the choice of Arens product in their biduals.

Corollary \thecorCt.

Let $A$ be a $C^{*}$-algebra and let $X$ be a Banach space that does not contain an isomorphic copy of $c_{0}$. Then every representation of $A$ on $X$ has a canonical extension to a representation of $A^{**}$.

For unital representations of unital $C^{*}$-algebras, Section 2 has implicitly been obtained by Spain, [Spa15, Section 9]. However, for our applications in Section 3, we have to consider nonunital Banach algebras and nonunital representations.

3. Projections onto the essential space of a representation

Let $X$ be a Banach space, and let $X_{0}\subseteq X$ be a closed subspace. Recall that a projection from $X$ onto $X_{0}$ is an idempotent operator on $X$ with image $X_{0}$. For $\lambda\geq 1$, we say that $X_{0}$ is $\lambda$-complemented in $X$ if there exists a projection from $X$ onto $X_{0}$ of norm at most $\lambda$. The space $X_{0}$ is said to be complemented if it is $\lambda$-complemented for some $\lambda$. In this section, we study the following:

Problem \thepbmCt.

Let $A$ be a Banach algebra, and let $X$ be a Banach space. Under which assumptions on $A$ and $X$ does it follow that the essential subspace of every representation $A\to{\mathcal{L}}(X)$ is complemented? Moreover, if such a projection exists, can its norm be estimated?

Of course, any solution to Section 3 depends crucially on $A$ and $X$. The better behaved $A$ is, the fewer assumptions we need to put on $X$, and conversely. The extreme points for this tradeoff are given by the case that $X$ is isomorphic to a Hilbert space (where every subspace is 1-complemented), and on the other end that $A$ has a left unit (where the image of the left unit is the desired projection).

The main result of this section is Section 3, which provides a positive solution to Section 3 for the case that $A$ has a bounded left approximate identity and every operator $A\to X$ is weakly compact.

The following example shows that every Banach space which is not a Hilbert space admits a representation of some Banach algebra whose essential space is not complemented.

Example \theexaCt.

Let $X$ be a Banach space not isomorphic to a Hilbert space. By the Lindenstrauss-Tzafriri theorem, [LT71], we can choose a closed subspace $X_{0}$ that is not complemented in $X$. Consider ${\mathcal{L}}(X,X_{0})$, which we identify with the space of operators $a\in{\mathcal{L}}(X)$ satisfying $a(X)\subseteq X_{0}$. Then ${\mathcal{L}}(X,X_{0})$ is a closed subalgebra of ${\mathcal{L}}(X)$. Then the inclusion map is an isometric representation $\varphi\colon{\mathcal{L}}(X,X_{0})\to{\mathcal{L}}(X)$, whose essential space is easily seen to be exactly $X_{0}$.

Analogously, one may ask what happens if the Banach algebra is fixed but the Banach space is allowed to vary:

Question \theqstCt.

Let $A$ be a Banach algebra without left unit. Does there exist a representation $\varphi\colon A\to{\mathcal{L}}(X)$ on some Banach space $X$ such that the essential space of $\varphi$ is not complemented?

Example \theexaCt.

Let $\varphi\colon c_{0}\to{\mathcal{L}}(\ell_{\infty})$ be the canonical isometric representation given by pointwise multiplication. It is easy to show that the essential space of $\varphi$ is exactly the subspace $c_{0}$ in $\ell_{\infty}$. However, Phillips’ theorem states that $c_{0}$ is not complemented in $\ell_{\infty}$. (See [Whi66] for a simple proof of Phillips’ theorem.) This provides a positive answer to Section 3 for $A=c_{0}$.

\thepgrCt**.**

We can generalize Section 3 to many other nonunital $C^{*}$-algebras, as follows. Taylor showed in [Tay72, Corollary 3.7] that a nonunital $C^{*}$-algebra $A$ is uncomplemented in its multiplier algebra $M(A)$ if $A$ has a well-behaved approximate identity. (See [Tay72] for the definition.) By [Tay72, Proposition 3.1], it follows that every non-unital, $\sigma$-unital $C^{*}$-algebra is uncomplemented in its multiplier algebra. In particular, this gives a positive answer to Section 3 for every separable $C^{*}$-algebra.

In general, however, a $C^{*}$-algebra need not be uncomplemented in its multiplier algebra. Indeed, there exist non-unital $C^{*}$-algebras whose multiplier algebras agree with their minimal unitizations, and every $C^{*}$-algebra is complemented in its minimal unitization. For examples and a further discussion, we refer to [GK16].

\thepgrCt**.**

Let $A$ be a Banach algebra. Given a bounded left approximate identity $(e_{j})_{j\in J}$ in $A$, let $e$ be any weak∗ cluster point of the net $(\kappa_{A}(e_{j}))_{j\in J}$ in $A^{**}$. Then $e\lozenge a=a$ for every $a\in A$. Since the right Arens product is weak∗ continuous in the second variable, we deduce that $e\lozenge S=S$ for all $S\in A^{**}$. Thus, $e$ is a left unit for the right Arens products. The converse also holds, and hence $A$ has a bounded left approximate identity (with bound $\lambda$) if and only if $(A^{**},\lozenge)$ has a left unit (with norm at most $\lambda$); see [Pal94, Proposition 5.1.8, p.527].

Lemma 3.1**.**

Let $A$ be a Banach algebra, let $X$ be a Banach space, and let $\varphi\colon A\to{\mathcal{L}}(X)$ be a representation. Set $\tilde{\varphi}=\alpha_{X}\circ\varphi^{**}\colon A^{**}\to{\mathcal{L}}(X,X^{**})$. Let $e\in A^{**}$ be a left unit for $(A^{**},\lozenge)$, and set $p=\tilde{\varphi}(e)$. Then $p$ is an idempotent in ${\mathcal{L}}(X,X^{**})$ with $\|p\|\leq\|e\|\|\varphi\|$ and such that $p\tilde{\varphi}(S)=\tilde{\varphi}(S)$ for all $S\in A^{**}$.

Proof.

By Section 2, we have $\|\tilde{\varphi}\|=\|\varphi\|$, which implies that

[TABLE]

Since $\tilde{\varphi}$ is multiplicative for the right Arens product by Section 2, we deduce that $p^{2}=\tilde{\varphi}(e\lozenge e)=\tilde{\varphi}(e)=p$. Finally, for every $S\in A^{**}$, we have

[TABLE]

The following is a (partial) solution to Section 3.

Theorem \thethmCt.

Let $A$ be a Banach algebra with a bounded left approximate identity with bound $\lambda$, let $X$ be a Banach space such that every operator $A\to X$ is weakly compact, and let $\varphi\colon A\to{\mathcal{L}}(X)$ be a representation. Then the essential space of $\varphi$ is $\lambda\|\varphi\|$-complemented in $X$.

Proof.

Set $\tilde{\varphi}=\alpha_{X}\circ\varphi^{**}\colon A^{**}\to{\mathcal{L}}(X,X^{**})$, as in Section 2. It follows from Section 2 that the image of $\tilde{\varphi}$ is contained in ${\mathcal{L}}(X)$, and that $X_{\varphi}=X_{\widetilde{\varphi}}$. As noted in Section 3, there exists a left unit $e$ for $(A^{**},\lozenge)$ with $\|e\|\leq\lambda$. Set $p=\tilde{\varphi}(e)\in{\mathcal{L}}(X)$. Then $p$ is an idempotent and $\|p\|\leq\lambda\|\varphi\|$ by Lemma 3.1. The range of $p$ is contained in the essential space of $\tilde{\varphi}$. Conversely, it follows from Lemma 3.1 that $p$ acts as the identity on the essential space of $\tilde{\varphi}$. Thus, $p$ is the desired projection onto the essential space of $\varphi$. ∎

Corollary \thecorCt.

Let $A$ be a Banach algebra with a contractive left approximate identity, let $X$ be a reflexive Banach space, and let $\varphi\colon A\to{\mathcal{L}}(X)$ be a contractive representation. Then the essential space of $\varphi$ is $1$-complemented in $X$.

Proof.

Since $X$ is reflexive, every operator $A\to X$ is weakly compact. Thus, we may apply Section 3. ∎

Corollary \thecorCt.

Let $A$ be a $C^{*}$-algebra, let $X$ be a Banach space that does not contain an isomorphic copy of $c_{0}$, and let $\varphi\colon A\to{\mathcal{L}}(X)$ be a contractive representation. Then the essential space of $\varphi$ is $1$-complemented in $X$.

Proof.

The assumptions imply that every operator $A\to X$ is weakly compact; see Section 2. Moreover, every $C^{*}$-algebra has a contractive approximate identity. Thus, we may apply Section 3 to deduce the statement. ∎

4. Representations of multiplier algebras

Throughout this section, we will use the notion of (left) multiplier algebras; see, for example, [Joh64] and [Daw10]. It is well-known that a nondegenerate representation of a Banach algebra $A$ with bounded approximate identity can be extended to a representation of the (left) multiplier algebra $M_{l}(A)$. For our purposes, we need to show that a nondegenerate isometric representation of $A$ can be extended to an unital, isometric representation of $M_{l}(A)$. This result is certainly also well-known, but we could not locate a reference. We therefore include an argument.

Theorem \thethmCt.

Let $X$ be a Banach space, let $A$ be a Banach algebra with a contractive, left approximate identity, and let $\varphi\colon A\to{\mathcal{L}}(X)$ be a contractive, nondegenerate representation. Then there is a unique contractive, unital representation $\widetilde{\varphi}\colon M_{l}(A)\to{\mathcal{L}}(X)$ such that the following diagram commutes:

[TABLE]

Moreover, for every $a\in A$, $x\in X$ and $T\in M_{l}(A)$, we have

[TABLE]

Furthermore, if $\varphi$ is isometric, then so is $\widetilde{\varphi}$.

Proof.

Let $T\in M_{l}(A)$. We claim that (1) defines a map $\widetilde{\varphi}(T)\colon X\to X$. By the Cohen-Hewitt factorization theorem, we have $X=\varphi(A)X$. To show that $\widetilde{\varphi}(T)$ is well-defined let $a,b\in A$ and $x,y\in X$ satisfy $\varphi(a)(x)=\varphi(b)(y)$. Let $(e_{j})_{j\in J}$ be a contractive, left approximate identity in $A$. Using that $T$ is continuous at the first and last step, and that $T$ is a left multiplier at the second step, we obtain that

[TABLE]

Analogously, we have

[TABLE]

Thus $\varphi(T(a))(x)=\varphi(T(b))(y)$, and $\widetilde{T}$ is well-defined.

We claim that $\widetilde{\varphi}$ is a unital representation satisfying $\widetilde{\varphi}\circ L=\varphi$. Given $S,T\in M_{l}(A)$, $a\in A$ and $x\in X$, we have

[TABLE]

and thus $\widetilde{\varphi}(ST)=\widetilde{\varphi}(S)\widetilde{\varphi}(T)$. Similarly one shows that $\widetilde{\varphi}({\mathrm{id}}_{A})={\mathrm{id}}_{E}$. Further,

[TABLE]

for all $b\in A$, which implies that $\widetilde{\varphi}(L_{a})=\varphi(a)$. Thus, $\widetilde{\varphi}\circ L=\varphi$, as desired.

We claim that $\widetilde{\varphi}$ is conctractive. Given $T\in M_{l}(A)$, we use the Cohen-Hewitt factorization theorem at the second step, to get

[TABLE]

as desired. Finally, when $\varphi$ is isometric, then $\widetilde{\varphi}$ is also isometric:

[TABLE]

Remark \thermkCt.

If $A$ has a contractive, approximate identity, then the analog of Section 4 holds with the multiplier algebra in place of the left multiplier algebra, since in this case $M(A)$ can be identified with a closed subalgebra of $M_{l}(A)$.

Let $p\in[1,\infty)$. Recall that a Banach space is called an $SL^{p}$-space ($QL^{p}$-space, $QSL^{p}$-space) if it is isometrically isomorphic to a subspace (respectively, a quotient, a quotient of a subspace) of an $L^{p}$-space. In Section 4, we consider classes of reflexive Banach spaces that are closed under passing to $1$-complemented subspaces. By [Tza69, Theorem 6], every $1$-complemented subspace of an $L^{p}$-space is again an $L^{p}$-space. It follows that for each $p$, the class of $L^{p}$-spaces (of $QL^{p}$-spaces, of $SL^{p}$-spaces, of $QSL^{p}$-spaces) is closed under passing to $1$-complemented subspaces. For an overview of classes of Banach spaces with this property, see [NR11].

Let $\mathcal{E}$ be a class of Banach spaces. A Banach algebra $A$ is called an $\mathcal{E}$-operator algebra if there exists an isometric representation of $A$ on some Banach space in $\mathcal{E}$. For instance, an $L^{p}$-operator algebra is a Banach algebra that admits an isometric representation on some $L^{p}$-space. Such algebras have been studied in [Phi12], [Phi13], [GT15a], [GT15b], [GT16b].

Theorem \thethmCt.

Let $\mathcal{E}$ be a class of reflexive Banach spaces that is closed under passing to $1$-complemented subspaces, and let $A$ be an $\mathcal{E}$-operator algebra with a contractive left approximate identity. Then there is a nondegenerate isometric representation of $A$ on a space in $\mathcal{E}$. It follows that the left multiplier algebra of $A$ has a unital, isometric representation on a space in $\mathcal{E}$.

If $A$ has a contractive approximate identity, then the same holds for $M(A)$.

Proof.

Choose $X\in\mathcal{E}$ and an isometric representation $\varphi\colon A\to{\mathcal{L}}(X)$. Let $X_{\varphi}$ denote the essential space of $\varphi$. By Section 3, $X_{\varphi}$ is $1$-complemented in $X$, so $X_{\varphi}\in\mathcal{E}$. The restriction of $\varphi$ to $X_{\varphi}$ is the desired nondegenerate representation. Applying Section 4, we obtain a unital, isometric representation of $M_{l}(A)$ on $X_{\varphi}$. If $A$ has a contractive approximate identity, then $M(A)$ can be identified with a unital, closed subalgebra of $M_{l}(A)$, which implies the statements for $M(A)$. ∎

Remark \thermkCt.

If $A$ is a $C^{*}$-algebra, then Section 4 also holds for the class of $L^{1}$-spaces; see Lemma 5.1.

5. Representations of $C^{*}$-algebras on $L^{p}$-spaces

The goal of this section is to show that a $C^{*}$-algebra is necessarily commutative if it has an isometric representation on an $L^{p}$-spaces for $p\in[1,\infty)\setminus\{2\}$; see Section 5. This result is a crucial ingredient to obtain the results in [GT16c], where it is shown that algebras of convolution operators on $L^{p}(G)$, for a nontrivial locally compact group $G$, are not presentable on an $L^{q}$-space unless $p=q$, or $p$ and $q$ are conjugate Hölder exponents, or $p=2$ and $G$ is commutative.

To main ingredients to prove Section 5 are: first, a reduction to the unital case using the results in Section 4; and second, Lamperti’s theorem [Lam58].

For $p>1$, the following result also follows from Section 4.

Lemma 5.1**.**

Let $p\in[1,\infty)$, and let $A$ be a $C^{*}$-algebra with an isometric representation on an $L^{p}$-space. Then the multiplier algebra of $A$ has a unital, isometric representation on an $L^{p}$-space.

Proof.

Choose an $L^{p}$-space $X$ and an isometric representation $\varphi\colon A\to{\mathcal{L}}(X)$. As observed in Section 2, $X$ does not contain an isomorphic copy of $c_{0}$. It follows from Section 3, that $X_{\varphi}$ is $1$-complemented in $X$. We may then argue as in the proof of Section 4 to deduce that $X_{\varphi}$ is an $L^{p}$-space and that the restriction of $\varphi$ to $X_{\varphi}$ extends to a unital isometric representation of $M(A)$ on $X_{\varphi}$. ∎

Given a unital Banach algebra $A$, recall that $a\in A$ is said to be hermitian if $\|\exp(ita)\|=1$ for every $t\in{\mathbb{R}}$. (Equivalently, $\|\exp(ita)\|\leq 1$ for every $t\in{\mathbb{R}}$.) An element of a unital $C^{*}$-algebra is hermitian if and only if it is self-adjoint.

Let $A$ and $B$ be unital Banach algebras, let $\varphi\colon A\to B$ be a multiplicative operator, and let $a\in A$ be hermitian. If $\varphi$ is unital and contractive, then $\varphi(a)$ is hermitian in $B$. Indeed, using that $\varphi$ is unital, we have $\exp(it\varphi(a))=\varphi(\exp(ita))$ for every $t$. Further, using that $\varphi$ is contractive, we obtain that

[TABLE]

as desired. If $\varphi$ is not unital or not contractive, then $\varphi(a)$ need not be hermitian.

Lemma 5.2**.**

Let $p\in[1,\infty)\setminus\{2\}$, let $(X,\mu)$ be a complete, $\sigma$-finite measure space, and let $a\in{\mathcal{L}}(L^{p}(\mu))$. Then $a$ is hermitian if and only if $a$ is the multiplication operator corresponding to some bounded, measurable function $X\to{\mathbb{R}}$. In particular, all hermitian operators on $L^{p}(\mu)$ commute.

Proof.

Assume that $a$ is given by multiplication with a bounded, measurable function $h\colon X\to{\mathbb{R}}$. We need to show that $\|\exp(ita)\|\leq 1$ for every $t\in{\mathbb{R}}$. Let $t\in{\mathbb{R}}$. Then the operator $\exp(ita)$ is given by multiplication with the function $\exp(ith)$. For each $x\in X$, we have $\exp(ith)(x)\in S^{1}$, and therefore $\|\exp(ith)\|_{\infty}\leq 1$. It follows that $\|\exp(ita)\|\leq 1$, as desired.

To show the converse implication, assume that $a$ is hermitian. By rescaling, if necessary, we may assume that $\|a\|\leq\tfrac{\pi}{2}$. Let ${\mathrm{Isom}}(L^{p}(\mu))$ denote the set of surjective isometric operators $L^{p}(\mu)\to L^{p}(\mu)$. For $t\in{\mathbb{R}}$, set $u_{t}=\exp(ita)$. Since $a$ is hermitian, we have $\|u_{t}\|\leq 1$ for every $t$. Moreover, we have $u_{t}u_{-t}=u_{-t}u_{t}={\mathrm{id}}$, which implies that $u_{t}$ belongs to ${\mathrm{Isom}}(L^{p}(\mu))$ for every $t$. Moreover, the resulting map $u\colon[0,1]\to{\mathrm{Isom}}(L^{p}(\mu))$ is norm-continuous.

By Lamperti’s theorem (in the form given in Theorem 4.4 in [GT15a]; see also [Lam58] for the original statement), for every $v\in{\mathrm{Isom}}(L^{p}(\mu))$ there exists a measurable function $h_{v}\colon X\to S^{1}$ and a measure class preserving automorphism $T_{v}\colon X\to X$ (see Definition 4.1 in [GT15a]) such that

[TABLE]

for all $\xi\in L^{p}(\mu)$ and $\mu$-almost every $x\in X$. (The transformation $T_{v}$ is called the spatial realization of $v$ in [Phi12].)

We may apply Lamperti’s theorem to $u_{t}$, for each $t\in[0,1]$. By Lemma 6.22 in [Phi12], the spatial realizations of $u_{0}$ and $u_{1}$ agree. Since $u_{0}$ is the identity map on $L^{p}(\mu)$, its spatial realization is the identity on $X$. Thus, there exists a measurable function $h\colon X\to S^{1}$ such that $u_{1}$ is given by multiplication with $h$.

Set $T=\{it:t\in[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\}$, and set $P=\{z\in S^{1}:{\mathrm{Re}}(z)\geq 0\}$. The restriction of the exponential map to $T$ is a bijection onto $P$. We let $\log\colon P\to T$ denote the inverse map, which is contained in a branch of the logarithm and is therefore analytic on a neighborhood of $P$.

Let $\sigma(a)$ denote the spectrum of $a$. Since $\|a\|\leq\tfrac{\pi}{2}$, the spectrum of $ia$ is contained in $T$, and consequently $\sigma(u_{1})\subseteq P$. Thus, we may apply (analytic) functional calculus to $u_{1}$ and obtain that $ia=\log(u_{1})$. Recall that $u_{1}$ is given by multiplication with the function $h\colon X\to S^{1}$. After changing $h$ on a null-set, we may assume that $h$ takes image in $P$. It follows that $ia$ is given by multiplication by the function $\log(h)$, which takes values in $T$. This implies that $a$ is given by multiplication by $-i\log(h)$, which is a bounded, measurable real-valued function, as desired. ∎

Theorem \thethmCt.

Let $A$ be a $C^{*}$-algebra and let $p\in[1,\infty)\setminus\{2\}$. Then $A$ can be isometrically represented on an $L^{p}$-space if and only if $A$ is commutative.

Proof.

Assume that $A$ is commutative. By the Gelfand representation theorem, there exists a locally compact Hausdorff space $X$ such that $A\cong C_{0}(X)$. By choosing a suitable family of Borel probability measures on $X$, it is easy to construct a faithful representation of $C_{0}(X)$ on an $L^{p}$-space, via a direct sum of representations by multiplication operators. We omit the details.

To show the converse, assume that $A$ has an isometric representation on an $L^{p}$-space. By Lemma 5.1, there is a unital, isometric representation of the multiplier algebra $M(A)$ on some $L^{p}$-space. Since $A$ is commutative if (and only if) $M(A)$ is commutative, we can assume that $A$ and its representation are unital. Since any element in a $C^{*}$-algebra is a linear combination of two self-adjoint elements, it is enough to show that self-adjoint elements in $A$ commute.

Let $a,b\in A_{\mathrm{sa}}$. Since the unital sub-$C^{*}$-algebra generated by $a$ and $b$, is a separable unital $C^{*}$-algebra, we may assume that $A$ itself is separable. In this case, by [Phi13, Proposition 1.25], there exist a separable $L^{p}$-space $E$ and a nondegenerate (and hence automatically unital) isometric representation $\varphi\colon A\to{\mathcal{L}}(E)$. Since every separable $L^{p}$-space is isometrically isomorphic to $L^{p}(\mu)$ for some complete, $\sigma$-finite measure space $\mu$, and since $\varphi(a)$ and $\varphi(b)$ are hermitian, we deduce from Lemma 5.2 that $\varphi(a)$ and $\varphi(b)$ commute. Since $\varphi$ is injective, the result follows. ∎

One important ingredient in the proof of Section 5 was that all hermitian operators on $L^{p}(\mu)$ commute, for a complete, $\sigma$-finite measure space $\mu$ and $p\in[1,\infty)\setminus\{2\}$. We remark that there are other Banach spaces $X$ such that all hermitian operators on $X$ commute. There are even Banach spaces $X$ such that every hermitian operator on $X$ is a multiple of the identity operator; see [BS74, Section 3]. For such $X$, there is no isometric, nondegenerate representation of any nonzero $C^{*}$-algebra $A\neq{\mathbb{C}}$ on $X$.

Remark \thermkCt.

Section 5 should be contrasted with the fact that every $C^{*}$-algebra can be isometrically represented on a noncommutative $L^{p}$-space To prove this, it is enough to show it for ${\mathcal{L}}(\mathcal{H})$ for a Hilbert space $H$. Given $p\in[1,\infty)$, consider the associated noncommutative $L^{p}$-space $\mathcal{S}_{p}(H)$ of Schatten-$p$ class operators. Using that $\mathcal{S}_{p}(H)$ is a (nonclosed) ideal in ${\mathcal{L}}(H)$, we may define $\varphi\colon{\mathcal{L}}(H)\to{\mathcal{L}}(\mathcal{S}_{p}(H))$ by $\varphi(a)b=ab$, for $a\in{\mathcal{L}}(H)$ and $b\in\mathcal{S}_{p}(H)$. It is well-known that $\varphi$ is an isometric representation.

6. Universal completions for representations on $L^{p}$-spaces

\thepgrCt**.**

Let $A$ be a Banach algebra, and let $\mathcal{E}$ be a class of Banach spaces. We let $\operatorname{Rep}_{\mathcal{E}}(A)$ denote the class of all non-degenerate, contractive representations $A\to{\mathcal{L}}(X)$, for $X$ ranging over Banach spaces in $\mathcal{E}$. Given $a\in A$, we set

[TABLE]

We let $F_{\mathcal{E}}(A)$ denote the Hausdorff completion of $A$ with respect to the semi-norm $\|\cdot\|_{\mathcal{E}}$. Multiplication on $A$ extends to $F_{\mathcal{E}}(A)$ and turns it into a Banach algebra.

In the next result, we show that in the definition of $F_{\mathcal{E}}(A)$, one can sometimes consider all representations on $\mathcal{E}$, and not just the nondegenerate ones.

Theorem \thethmCt.

Let $\mathcal{E}$ be a class of reflexive Banach spaces that is closed under passing to $1$-complemented subspaces, and let $A$ be a Banach algebra with a contractive left approximate identity, and let $a\in A$. Then

[TABLE]

Proof.

Let $K$ denote the supremum of $\|\varphi(a)\|$ for all contractive representations $\varphi\colon A\to{\mathcal{L}}(X)$ with $X\in\mathcal{E}$. We clearly have $\|a\|_{\mathcal{E}}\leq K$. To show the converse inequality, let $X\in\mathcal{E}$ and let $\varphi\colon A\to{\mathcal{L}}(X)$ be a contractive representation. We need to verify that $\|\varphi(a)\|\leq\|a\|_{\mathcal{E}}$. It follows from Section 3 that $X_{\varphi}$ is $1$-complemented in $X$, and hence $X_{\varphi}$ belongs to $\mathcal{E}$. Let $\varphi_{0}$ denote the restriction of $\varphi$ to $X_{\varphi}$. Note that $\|\varphi(b)\|=\|\varphi_{0}(b)\|$ for every $b\in A$. Thus, $\varphi_{0}$ is a nondegenerate, contractive representation of $A$ on a space in $\mathcal{E}$. We conclude that

[TABLE]

Let $G$ be a locally compact group. We equip $G$ with a fixed left invariant Haar measure, and we let $L^{1}(G)$ denote the corresponding group algebra of integrable functions, with product given by convolution. Note that $L^{1}(G)$ is a Banach algebra with a contractive, approximate identity.

Let $X$ be a Banach space, and let ${\mathrm{Isom}}(X)$ denote the group of surjective isometric operators $X\to X$. We equip ${\mathrm{Isom}}(X)$ with the strong operator topology. An isometric representation of $G$ on $X$ is a continuous group homomorphism $G\to{\mathrm{Isom}}(X)$. Every isometric representation of $G$ on $X$ can be integrated to obtain a nondegenerate contractive representation $L^{1}(G)\to{\mathcal{L}}(X)$. It is well-known that this induces a bijection between isometric representations of $G$ on $X$ and nondegenerate, contractive representations $L^{1}(G)\to{\mathcal{L}}(X)$.

Given a class $\mathcal{E}$ of Banach spaces, we denote $F_{\mathcal{E}}(L^{1}(G))$ by $F_{\mathcal{E}}(G)$. The Banach algebra $F_{\mathcal{E}}(G)$ captures the representation theory of $G$ on Banach spaces in $\mathcal{E}$. Given $p\in[1,\infty)$, let $L^{p}$ denote the class of $L^{p}$-spaces. We set $F^{p}(G)=F_{L^{p}}(G)$, and we call $F^{p}(G)$ the universal group $L^{p}$-operator algebra of $G$. The Banach algebras $F^{p}(G)$ have been studied in [GT15b].

Corollary \thecorCt.

Let $p\in(1,\infty)$, and let $G$ be a locally compact group. Then for each $f\in L^{1}(G)$, the norm $\|f\|_{L^{p}}$ agrees with the supremum of $\|\varphi(f)\|$ for all (not necessarily nondegenerate) contractive representations $\varphi\colon L^{1}(G)\to{\mathcal{L}}(X)$ on some $L^{p}$-space $X$. Consequently, $F^{p}(G)$ is universal for all contractive representations of $L^{1}(G)$ on $L^{p}$-spaces.

Analogous statements hold for the universal group $SL^{p}$-operator algebra (group $QL^{p}$-operator algebra, group $QSL^{p}$-operator algebra).

Section 6 is a crucial ingredient in [GT16c], where it is used to reduce the question of representability of certain Banach algebras of convolution on $L^{q}$-spaces to a question about nondegenerate representations. In particular, for every nontrivial, locally compact group $G$, it follows that $F^{p}(G)$ and $F^{q}(G)$ are not isometrically isomorphic as Banach algebras for $p,q\in[1,2]$ with $p\neq q$.