A Banach algebra encoding quantum group duality

Abstract: We introduce and study a new Banach algebra structure on the trace-zero subspace $\mathcal{T}(L^2(\mathbb{G}))_0$ of trace class operators for any locally compact quantum group $\mathbb{G}$; it is defined through a mixed Lie-type product of the two dual products on $\mathcal{T}(L^2(\mathbb{G}))$ arising from the canonical extensions of the co-products of $\mathbb{G}$ and $\widehat{\mathbb{G}}$. The surprising fact that this new product is indeed associative stems precisely from the duality of the latter two products. This, in particular, gives new faithful associative products on trace-zero matrices in $M_d(\mathbb{C})$. After establishing some basic properties, we show that the single algebra $\mathcal{T}(L^2(\mathbb{G}))_0$ captures simultaneous properties of $\mathbb{G}$ and $\widehat{\mathbb{G}}$, is faithful for a large class of quantum groups, and encodes both $M^r_{cb}(L^1(\mathbb{G}))$ and $M^r_{cb}(L^1(\widehat{\mathbb{G}}))$ as left, respectively right, completely bounded module maps on $\mathcal{T}(L^2(\mathbb{G}))$. We finish by exhibiting an analogous product on the trace-zero nuclear operators $\mathcal{N}(L^p(G))_0$ for a locally compact group $G$ and $p\in(1,\infty)$. Building on [7], our work suggests an approach for developing an $L^p$-version of locally compact quantum group theory.

• The paper constructs associative, completely contractive Lie- and Jordan-type products on the trace-zero trace class using the canonical products of a locally compact quantum group and its dual.
• The resulting algebra has a bounded approximate identity exactly when both the quantum group and its dual are co-amenable, linking Banach algebra structure to a fundamental quantum-group property.
• Under co-amenability, completely bounded multiplier algebras of the quantum group and its dual appear as left and right completely bounded module maps over this single algebra, with an analogous construction for nuclear operators on L^p(G).

Overview

This paper, by Crann and Neufang (2602.19589), constructs a new Banach algebra structure on the trace-zero subspace $T_0$ of the trace class operators $T = B(L^2(G))_*$ associated to an arbitrary locally compact quantum group $\mathbb{G}$. The construction mixes the two canonical products on $T$: the product $\star$ induced by the pre-adjoint of the right co-product extension of $\Gamma$, and its dual counterpart $\starhat$ arising from $\widehat{\mathbb{G}}$. The mixed product is

$\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$

a Lie-type combination of two associative products. The central observation is that this product is nevertheless associative on $T_0$, and that associativity is a direct consequence of the duality relations between $T = B(L^2(G))_*$0 and $T = B(L^2(G))_*$1 established by Kalantar–Neufang. A companion Jordan-type product $T = B(L^2(G))_*$2 (with a plus sign) enjoys the same properties.

The paper's thesis is that the single algebra $T = B(L^2(G))_*$3 captures simultaneous properties of both $T = B(L^2(G))_*$4 and $T = B(L^2(G))_*$5, whereas each individual product on $T = B(L^2(G))_*$6 reflects only one of the two. This is made precise through a bounded approximate identity characterization (co-amenability of both $T = B(L^2(G))_*$7 and $T = B(L^2(G))_*$8) and through representations of the completely bounded multiplier algebras $T = B(L^2(G))_*$9 and $\mathbb{G}$0 as left and right completely bounded module maps over $\mathbb{G}$1.

The dual products and their compatibility

The right fundamental unitary $\mathbb{G}$2 implements an extended co-multiplication $\mathbb{G}$3 on $\mathbb{G}$4, whose pre-adjoint yields a completely contractive, associative product $\mathbb{G}$5 on $\mathbb{G}$6 with $\mathbb{G}$7. The analogous construction for $\mathbb{G}$8 gives $\mathbb{G}$9. Each product is degenerate: $T$0 whenever $T$1, so neither alone sees the full structure of $T$2.

The key structural input is Proposition 3.2 of Kalantar–Neufang, which states that for all $T$3,

$T$4

These identities are precisely what makes the mixed product associative once the middle factor lies in $T$5; in fact, only the middle element needs trace zero, so $T$6 becomes an operator bimodule over $T$7 in the canonical way. The authors distill this into an abstract result (Proposition 4.3): any Banach space carrying two associative products satisfying these two mutual-commutativity conditions supports associative Lie/Jordan-type mixed products. This abstraction is what later enables the $T$8 analogue.

A concrete formula relating the two products follows from a new identity between dual fundamental unitaries: there exists a unitary $T$9 with $\star$0, whence

$\star$1

For unimodular Kac algebras with tracial underlying von Neumann algebra, $\star$2; the authors note explicitly that the stronger identity could hold more generally but do not establish it.

Basic structural results

Several consequences follow directly. When $\star$3 is finite of dimension $\star$4, $\star$5 identifies with $\star$6, and Theorem 4.1 produces new associative completely contractive products on it; for finite groups the product admits an explicit entry-wise formula mixing "left" and "right" diagonals of the matrices, with an explicit identity element $\star$7.

The quotient maps behave well: the restriction $\star$8 is a quotient map and $\star$9 is multiplicative for $\Gamma$0; symmetrically, $\Gamma$1 is multiplicative up to sign onto $\Gamma$2. Consequently, abelianness of $\Gamma$3 forces abelianness of both augmentation ideals, and under mild hypotheses (co-amenability of either $\Gamma$4 or $\Gamma$5, or factorization of either augmentation ideal), Lemma 5.5 upgrades abelianness of an augmentation ideal to abelianness of the full convolution algebra. The conclusion is sharp: if $\Gamma$6 under any of these conditions, $\Gamma$7 is nonabelian — in particular, for every locally compact group with at least two elements. The proof in the commutative case is a genuine geometric argument using approximate identities supported near distinct points and the intersection relation $\Gamma$8.

An identity worth noting: for $\Gamma$9 one has $\starhat$0, which coincides with the shuffle identity of Ebrahimi-Fard–Patras; the authors credit R. Speicher for this observation.

Bounded approximate identities

The main structural theorem states:

Theorem. $\starhat$1 has a bounded approximate identity if and only if both $\starhat$2 and $\starhat$3 are co-amenable.

The forward direction uses nets of unit vectors $\starhat$4, $\starhat$5 furnished by co-amenability of $\starhat$6 and $\starhat$7 respectively, each satisfying convergence of $\starhat$8- and $\starhat$9-averages together with $\widehat{\mathbb{G}}$0-invariance. The rank-one differences $\widehat{\mathbb{G}}$1, scaled by $\widehat{\mathbb{G}}$2, then form a bounded approximate identity via iterated limits. The converse pushes a bounded approximate identity of $\widehat{\mathbb{G}}$3 through the quotient maps to obtain bounded approximate identities for both augmentation ideals $\widehat{\mathbb{G}}$4 and $\widehat{\mathbb{G}}$5, which by Lau's theorem entails amenability and co-amenability of both quantum groups.

Two corollaries sharpen this. First, $\widehat{\mathbb{G}}$6 is unital if and only if $\widehat{\mathbb{G}}$7 is finite. Second, when both quantum groups are co-amenable, the left module $\widehat{\mathbb{G}}$8 over $\widehat{\mathbb{G}}$9 is faithful if and only if $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$0 is not compact. The paper also records that amenability of $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$1 itself, and even computation of amenability constants for finite groups, remain open.

Completely bounded multipliers

It is known from Hu–Neufang–Ruan that $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$2 and $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$3 embed completely isometrically into $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$4 as left $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$5-module maps and left $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$6-module maps, respectively. The paper's second main theorem shows that under co-amenability of both quantum groups, these images coincide exactly with the left, respectively right, completely bounded module maps over $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$7:

$\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$8

as completely isometric anti-representations, with a one-dimensional correction term $\rho \ostar \tau := \tfrac{1}{2}(\rho \star \tau - \tau \starhat \rho),$9 in the compact case ($T_0$0 denoting the Haar state). The proof is the most involved part of the paper: given a left $T_0$1-module map $T_0$2, the module identity forces $T_0$3 to be a left $T_0$4-module map on $T_0$5; passing to adjoints and composing with a conditional expectation built from an invariant mean yields an element of $T_0$6, which by co-amenability equals $T_0$7 for some $T_0$8. Singular/normal decomposition arguments then isolate the $T_0$9 component, and the remaining scalar term is shown to vanish unless $T = B(L^2(G))_*$00 is compact, in which case a Powers–Størmer argument forces it to be a multiple of $T = B(L^2(G))_*$01.

Consequently, when $T = B(L^2(G))_*$02 and $T = B(L^2(G))_*$03 are co-amenable and neither is compact, $T = B(L^2(G))_*$04 and $T = B(L^2(G))_*$05 are completely isometric anti-isomorphisms onto the respective module map spaces. In other words, the multipliers of both $T = B(L^2(G))_*$06 and its dual are realized as module maps over a single Banach algebra. The authors note that relating the multipliers of $T = B(L^2(G))_*$07 itself to $T = B(L^2(G))_*$08 and $T = B(L^2(G))_*$09 is deferred to future work.

An $T = B(L^2(G))_*$10 analogue

The final section extends the construction beyond Hilbert space. For a locally compact group $T = B(L^2(G))_*$11 and $T = B(L^2(G))_*$12, Daws–Spronk's framework provides a fundamental isometry $T = B(L^2(G))_*$13 on $T = B(L^2(G))_*$14 inducing a co-product on $T = B(L^2(G))_*$15, and its dual $T = B(L^2(G))_*$16 induces a second associative product on the nuclear operators $T = B(L^2(G))_*$17. The crucial point is that the same commutation relations hold at the level of isometries:

$T = B(L^2(G))_*$18

verified by direct coordinate computation. Under the canonical trivariate embedding of nuclear operators, these yield exactly the two hypotheses of the abstract Proposition 4.3 on $T = B(L^2(G))_*$19, producing associative Banach algebra products

$T = B(L^2(G))_*$20

The authors present this as evidence that the commutation relation between dual fundamental isometries can serve as a pentagonal-type axiom in a putative $T = B(L^2(G))_*$21-version of locally compact quantum group theory, building on the $T = B(L^2(G))_*$22-operator algebra program of Phillips, Gardella–Thiel, and others. It should be stressed that this section concerns locally compact groups only; no general $T = B(L^2(G))_*$23 quantum group formalism is constructed here.

Limitations and open questions

The paper is candid about scope. The identity $T = B(L^2(G))_*$24 is proven only for unimodular tracial Kac algebras, and the general unitary $T = B(L^2(G))_*$25 is not identified. Faithfulness of $T = B(L^2(G))_*$26 is established only for a restricted class of quantum groups (via the co-amenability hypotheses in the multiplier theorem); no faithfulness criterion is given in general. Amenability of $T = B(L^2(G))_*$27, its amenability constants even for finite groups, homological properties such as flatness and projectivity, and the centre $T = B(L^2(G))_*$28 are all left open. The $T = B(L^2(G))_*$29 construction applies only to commutative quantum groups, and whether the mixed-product mechanism extends to genuinely noncommutative, non-Kac settings at $T = B(L^2(G))_*$30 remains unresolved.

Conclusion

The paper demonstrates that a Lie-type mixture of the two canonical trace-class products yields an associative, completely contractive Banach algebra on $T = B(L^2(G))_*$31 whose associativity is a manifestation of quantum group duality. The algebra simultaneously detects co-amenability of $T = B(L^2(G))_*$32 and $T = B(L^2(G))_*$33 through its approximate identity theory, realizes both completely bounded multiplier algebras as one-sided module maps over a common product, and admits an $T = B(L^2(G))_*$34 analogue for locally compact groups via the Daws–Spronk isometries. The open problems concerning amenability constants, the centre, and an axiomatic $T = B(L^2(G))_*$35 quantum group theory delineate the natural continuation of this work.