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L1 group algebras have a contractively bounded approximate identity

Statement

Assume AC. Let $G$ be an LCH group with a fixed left Haar measure $μ$ , and let $U$ be the directed set of identity neighbourhoods of $G$ ordered by reverse inclusion. Then there is a net $(e_{U})_{U \in U} \subseteq C_{c} (G)$ with $e_{U} \geq 0, supp e_{U} \subseteq U, ∥ e_{U} ∥_{1} = 1,$ such that $∥ e_{U} * f - f ∥_{1} \to 0$ and $∥ f * e_{U} - f ∥_{1} \to 0$ for every $f \in L^{1} (G)$ : for each $ϵ > 0$ there is $U_{0} \in U$ with $∥ e_{U} * f - f ∥_{1} < ϵ$ and $∥ f * e_{U} - f ∥_{1} < ϵ$ for every $U \subseteq U_{0}$ in $U$ . Such a net is a contractively bounded approximate identity: bounded by $1$ in norm, with two-sided convergence, and no assumption that $G$ be discrete or first countable.

Facts & Assumptions

Given: An LCH group $G$ with a fixed left Haar measure $μ$ , the algebra $L^{1} (G)$ with convolution $*$ and involution $*$ , the directed set $U$ of identity neighbourhoods, and AC.

[F1]

If $K \subseteq U$ with $K$ compact, $U$ open and $G$ LCH, then under Dependent Choice there is $f \in C_{c} (G)$ with $1_{K} \leq f \leq 1_{U}$ ; the construction in the cited proof gives $supp f \subseteq U$ (LCH Urysohn cutoff).

[F2]

AC implies Dependent Choice: a choice function on the family ${R [x] : x \in X}$ of a serial relation, composed with recursion (The recursion theorem), produces the required sequence (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, The Axiom of Choice).

[F3]

A left Haar measure is nonzero, positive on every nonempty open set, and finite on compact sets (Haar measure is positive on nonempty open sets and finite on compact sets, Left Haar integral and left Haar measure).

[F4]

$e \mapsto e$ is the identity of $G$ ; inversion $inv (x) = x^{- 1}$ is a homeomorphism, so $U^{- 1}$ is an identity neighbourhood whenever $U$ is, and $U \mapsto U^{- 1}$ is a bijection of $U$ preserving inclusions (Left and right translations and inversion in a topological group are homeomorphisms).

[F5]

$(f * g) (x) = \int_{G} f (y) g (y^{- 1} x) d μ (y)$ for $f, g \in C_{c} (G)$ , and $f * g \in C_{c} (G)$ under AC (Compactly supported convolution on a group, Convolution preserves compact support and is associative).

[F6]

For a continuous compactly supported kernel on a product of LCH spaces the iterated integrals commute, in the real and in the complex case (Compactly supported kernels admit commuting radon integrals).

[F7]

Convolution on $L^{1} (G)$ is the unique bilinear extension of the $C_{c}$ convolution with $∥ f * g ∥_{1} \leq ∥ f ∥_{1} ∥ g ∥_{1}$ , hence jointly continuous (Convolution on L1 of a locally compact group, Submultiplicativity of convolution in the L1 norm).

[F8]

The involution satisfies $(f * g)^{*} = g^{*} * f^{*}$ , $(f^{*})^{*} = f$ and $∥ f^{*} ∥_{1} = ∥ f ∥_{1}$ ; for real nonnegative $g \in C_{c} (G)$ one has $g^{*} (x) = Δ_{G} (x^{- 1}) g (x^{- 1}) \geq 0$ with $supp g^{*} = (supp g)^{- 1}$ (The L1 involution is isometric, involutive and reverses convolution, Involution on L1 of a locally compact group, Translations preserve compactly supported continuous functions).

[F9]

$C_{c} (G)$ is dense in $L^{1} (G)$ (Completeness of the complex Haar L1 and L2 spaces and density of Cc, Compact support, $C_{c} (X)$ , and $C_{0} (X)$ ).

[F10]

For $f \in C_{c} (G)$ and $ϵ > 0$ there is an identity neighbourhood $V$ with $∥ L_{g} f - f ∥_{1} < ϵ$ for every $g \in V$ , where $L_{g} f (x) = f (g^{- 1} x)$ (Strong continuity of left and modular right translations on L1 and L2).

[A1]

AC is assumed, in the choice-function form of the cited definition; it supplies both the cutoffs of [F1] through [F2] and the single selection of one cutoff for each identity neighbourhood in step 1.1 (The Axiom of Choice).

Proof

technique · direct

1.1

Construction of the net. Let $U \in U$ be an identity neighbourhood and choose an open identity neighbourhood $W \subseteq U$ . Applying [F1] with $K = {e} \subseteq W$ under the Dependent Choice of [F2] produces $f_{U} \in C_{c} (G)$ with $1_{{e}} \leq f_{U} \leq 1_{W}$ and, by the cited construction, $supp f_{U} \subseteq W \subseteq U$ ; in particular $f_{U} \geq 0$ and $f_{U} (e) = 1$ . Since $f_{U}$ is continuous with $f_{U} (e) = 1$ , it exceeds $1 / 2$ on a neighbourhood of $e$ , so $\int_{G} f_{U} d μ > 0$ by [F3]; put $e_{U} : = f_{U} / \int_{G} f_{U} d μ$ . Then $e_{U} \in C_{c} (G)$ , $e_{U} \geq 0$ , $supp e_{U} \subseteq U$ and $∥ e_{U} ∥_{1} = 1$ . Choosing one such $e_{U}$ for each $U \in U$ is a single application of [A1] to the family of nonempty sets of admissible normalised cutoffs, and the resulting family is indexed by the directed set $U$ ; this is the net whose properties are claimed.

A1F1F2F3

1.2

Left convergence for compactly supported $f$ . Let $f \in C_{c} (G)$ and let $U \in U$ . Since $∥ e_{U} ∥_{1} = 1$ and $e_{U} \geq 0$ , the $C_{c}$ formula [F5] gives $(e_{U} * f) (x) - f (x) = \int_{G} e_{U} (y) (f (y^{- 1} x) - f (x)) d μ (y) = \int_{G} e_{U} (y) (L_{y} f - f) (x) d μ (y)$ for the translate $L_{y} f$ of [F10]. The kernel $(y, x) \mapsto e_{U} (y) ∣ (L_{y} f - f) (x) ∣$ is continuous with compact support: $y$ ranges in $supp e_{U} \subseteq U$ and $L_{y} f - f$ is supported in a compact set depending on $supp e_{U}$ and $supp f$ , as in [F5]. So [F6] applies to the complex kernel and, with $\int_{G} e_{U} d μ = 1$ , $∥ e_{U} * f - f ∥_{1} \leq \int_{G} ⁣ ⁣ \int_{G} e_{U} (y) ∣ (L_{y} f - f) (x) ∣ d μ (x) d μ (y) \leq \sup_{y \in U} ∥ L_{y} f - f ∥_{1} .$ By [F10], given $ϵ > 0$ there is $U_{0} \in U$ with $\sup_{y \in U} ∥ L_{y} f - f ∥_{1} < ϵ$ whenever $U \subseteq U_{0}$ ; hence $∥ e_{U} * f - f ∥_{1} \to 0$ along $U$ .

F5F6F10step 1.1

2.1

Left convergence for all of $L^{1} (G)$ . Let $f \in L^{1} (G)$ and $ϵ > 0$ . By [F9] choose $h \in C_{c} (G)$ with $∥ f - h ∥_{1} < ϵ / 3$ , and by step 1.2 choose $U_{0}$ with $∥ e_{U} * h - h ∥_{1} < ϵ / 3$ for all $U \subseteq U_{0}$ . For such $U$ , bilinearity and the norm bound of [F7] give $∥ e_{U} * f - f ∥_{1} \leq ∥ e_{U} * (f - h) ∥_{1} + ∥ e_{U} * h - h ∥_{1} + ∥ h - f ∥_{1} \leq 2 ∥ f - h ∥_{1} + ϵ / 3 < ϵ$ , since $∥ e_{U} ∥_{1} = 1$ .

F7F9step 1.1step 1.2

3.1

Right convergence. Let $f \in L^{1} (G)$ and $ϵ > 0$ . For every identity neighbourhood $U$ , [F8] gives $(f * e_{U})^{*} = e_{U}^{*} * f^{*}$ , with $e_{U}^{*} \in C_{c} (G)$ , $e_{U}^{*} \geq 0$ , $∥ e_{U}^{*} ∥_{1} = ∥ e_{U} ∥_{1} = 1$ and $supp e_{U}^{*} = (supp e_{U})^{- 1} \subseteq U^{- 1}$ . The calculation in step 1.2 applies to any nonnegative unit-mass $C_{c}$ kernel supported in a neighbourhood: for $h \in C_{c} (G)$ it gives $∥ e_{U}^{*} * h - h ∥_{1} \leq \sup_{y \in U^{- 1}} ∥ L_{y} h - h ∥_{1}$ . By [F10] choose an identity neighbourhood $V$ on which $∥ L_{y} h - h ∥_{1} < ϵ / 3$ , and put $U_{0} : = V^{- 1}$ ; then this bound is below $ϵ / 3$ for all $U \subseteq U_{0}$ . Given $f^{*} \in L^{1} (G)$ , first choose $h \in C_{c} (G)$ with $∥ f^{*} - h ∥_{1} < ϵ / 3$ by [F9]. The estimate of step 2.1, now using $e_{U}^{*}$ and [F7], yields $∥ e_{U}^{*} * f^{*} - f^{*} ∥_{1} < ϵ$ for all $U \subseteq U_{0}$ . Since the involution is isometric and involutive by [F8], $∥ f * e_{U} - f ∥_{1} = ∥ (f * e_{U})^{*} - f^{*} ∥_{1} < ϵ$ .

F4F7F8F9F10step 1.2step 2.1

4.1

The net $(e_{U})_{U \in U}$ of step 1.1 satisfies $e_{U} \geq 0$ , $supp e_{U} \subseteq U$ and $∥ e_{U} ∥_{1} = 1$ for every $U$ , and steps 2.1 and 3.1 show $∥ e_{U} * f - f ∥_{1} \to 0$ and $∥ f * e_{U} - f ∥_{1} \to 0$ for every $f \in L^{1} (G)$ along the directed set of identity neighbourhoods ordered by reverse inclusion. ∎

step 1.1step 2.1step 3.1

Remarks

Two-sidedness without discreteness. The net converges to the identity operator in the strong operator sense. If $G$ is discrete, ${e}$ is an identity neighbourhood and all terms beyond it equal $μ ({e})^{- 1} 1_{{e}}$ , the convolution unit. A unit need not exist in general, as characterised by The L1 group algebra has a unit exactly when the group is discrete.
Why the involution is needed. Step 3.1 is the only place where the modular factor enters, through the nonnegativity of $e_{U}^{*}$ and the identity $(f * e_{U})^{*} = e_{U}^{*} * f^{*}$ ; the left and right assertions of the statement are therefore not proved independently.
Choice cost. [A1] is used twice: through [F1] (Dependent Choice, via [F2]) to obtain the cutoffs, and once to select a normalised cutoff for each identity neighbourhood. The convergence estimates themselves are choice-free.

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Sources

Lynn Loomis, An Introduction to Abstract Harmonic Analysis, §§30–31 (standard reference, not scraped)
Bekka, de la Harpe and Valette, Kazhdan’s Property (T), Appendix A §§A.3–A.4 (standard reference, not scraped)