Equivariant Join and Fusion of Noncommutative Algebras

We translate the concept of the join of topological spaces to the language of $C^*$-algebras, replace the $C^*$-algebra of functions on the interval $[0,1]$ with evaluation maps at $0$ and $1$ by a unital $C^*$-algebra $C$ with appropriate two surjections, and introduce the notion of the fusion of unital $C^*$-algebras. An appropriate modification of this construction yields the fusion comodule algebra of a comodule algebra $P$ with the coacting Hopf algebra $H$. We prove that, if the comodule algebra $P$ is principal, then so is the fusion comodule algebra. When $C=C([0,1])$ and the two surjections are evaluation maps at $0$ and $1$, this result is a noncommutative-algebraic incarnation of the fact that, for a compact Hausdorff principal $G$-bundle $X$, the diagonal action of $G$ on the join $X*G$ is free.


Introduction
The join of topological spaces is a crucial concept in algebraic topology -it is used in the celebrated Milnor's construction of a universal principal bundle [6]. The goal of this paper is to provide a noncomutative-geometric generalization of the join of a Cartan compact Hausdorff principal bundle (no local triviality assumed) with its structure group. In particular, we generalize this way the n-fold join G * · · · * G of a compact Hausdorff topological group G, which is the first step in Milnor's construction.
To make this paper self-contained and to establish notation and terminology, we begin by recalling the basics of classical joins and principal comodule algebras. In the first section, we define the join C*-algebra of an arbitrary pair of unital C*-algebras. For commutative C*-algebras our construction recovers Milnor's definition in the case of compact Hausdorff topological spaces. In the second section, we deal with a join of a comodule algebra P with the coacting Hopf algebra H. To equip the join algebra P * H with an H-comodule algebra structure that is equivalent to the diagonal action in the classical setting, we provide a "gauged" algebraic version of the join C*-algebra. (A gauged C*-algebraic version is given in [1,3].) The main result of this paper is Theorem 4.3 concluding the principality of the join comodule algebra P * H from the principality of P .

Classical join construction
Let I = [0, 1] be the closed unit interval and let X be a topological space. The unreduced suspension ΣX of X is the quotient of I × X by the equivalence relation R S generated by Now take another topological space Y and, on the space I × X × Y , consider the equivalence relation R J given by The quotient space X * Y := (I × X × Y )/R J is called the join of X and Y . It resembles the unreduced suspension of X × Y , but with only X collapsed at 0, and only Y collapsed at 1.
In particular, if Y is a one-point space, the join X * Y is the cone CX of X. If Y is a two-point space with discrete topology, then the join X * Y is the unreduced suspension ΣX of X.
If G is a topological group acting continuously on X and Y from the right, then the diagonal right G-action on X × Y induces a continuous action on the join X * Y . Indeed, the diagonal action of G on I × X × Y factorizes to the quotient, so that the formula makes X * Y a right G-space. It is immediate that this continuous action is free if the G-actions on X and Y are free.
Consider CX × Y , X × CY and X × Y as G-spaces with the diagonal G-actions. Note that there is a continuous surjection are G-equivariantly homeomorphic to CX × Y and X × CY respectively. Thus X * Y is G-equivariantly homeomorphic with the gluing of CX × Y and X × CY over X × Y : If Y = G with the right action assumed to be the group multiplication, we can construct the join G-space X * Y in a different manner: at 0 we collapse X × G to G as before, and at 1 we One can also easily check that the formula If we further specify also X = G with the right action assumed to be the group multiplication, then the G-action on X * Y = G * G is automatically free. Furthermore, since the action of G on X * G is free whenever it is free on X, we conclude that the natural action on the iterated join of G with itself is also free. For instance, for G = Z/2Z we obtain a Z/2Z-equivariant identification (Z/2Z) * (n+1) ∼ = S n , where S n is the n-dimensional sphere with the antipodal action of Z/2Z.

Principal comodule algebras
Let H be a Hopf algebra with coproduct ∆, counit ε and bijective antipode S. Next, let ∆ P : P → P ⊗ H be a coaction making P a right H-comodule algebra. We shall frequently use the Heyneman-Sweedler notation (with the summation sign suppressed) for coproduct and coactions: Definition 2.1 ( [5]). Let P be a right comodule algebra over a Hopf algebra H with bijective antipode, and let be the coaction-invariant subalgebra. The comodule algebra P is called principal if the following conditions are satisfied: (1) the coaction of H is Hopf-Galois, that is, the map  (2) can P • = 1 ⊗ id, where can P : P ⊗ P p ⊗ q → (p ⊗ 1)∆ P (q) ∈ P ⊗ H.
We will use the Heyneman-Sweedler-type notation (h) =: (h) 1 ⊗ (h) 2 with the summation sign suppressed. One can easily prove (see [5, p. 599] and references therein) that a comodule algebra is principal if and only if it admits a strong connection. Here we need the following slight generalization of this fact: Lemma 2.3. Let H be a Hopf algebra with bijective antipode. Then a right H-comodule algebra P is principal if and only if it admits a (not nececessarily unital) linear map : H → P ⊗ P satisfying the bicolinearity and splitting conditions of Definition 2.2.
Proof. Assume that : H → P ⊗ P is a linear map satisfying the bicolinearity and splitting conditions. Let π B : P ⊗ P → P ⊗ B P be the canonical surjection. Define It follows immediately from the splitting property of that Applying id⊗ε to the splitting condition for , we obtain m• = ε, where m is the multiplication map on P . Combining it with the left colinearity of , we obtain: Finally, the bicolinearity of implies that the formula s(p) := p (0) (p (1) ) defines a left B-linear right H-colinear splitting of the multiplication map B ⊗ P → P . The reverse implication (principality ⇒ existence of a bicolinear with the splitting property) follows from Lemma 2.2 in [2].

Join of unital C*-algebras
Definition 3.1. Let A 1 and A 2 be unital C*-algebras. We call the unital C*-algebra the join C*-algebra of A 1 and A 2 . Here ⊗ min stands for the spatial (minimal) tensor product, and ev r : C([0, 1]) f → f (r) ∈ C is the evaluation map at r on the C*-algebra of continuous functions on the unit interval.
Note that, due to the fact that minimal tensor products preserve injections (e.g., see [8,Proposition 4.22] or [9, Section 1.3]), for any unital C*-algebras A 1 and A 2 the natural maps are injective. Observe also that, if A 1 := C(X) and A 2 := C(Y ) are the C*-algebras of continuous functions on compact Hausdorff spaces X and Y respectively, then (3.14) Remark 3.2. Let C be unital C*-algebra equipped with two C*-ideals J 1 and J 2 such that the direct sum π 1 ⊕ π 2 : C −→ C/J 1 ⊕ C/J 2 of the canonical surjections π i : C → C/J i , i ∈ {1, 2}, is surjective. In Definition 3.1, one can replace C([0, 1]) by C, π 1 and π 2 by ev 0 and ev 1 respectively, and A i , i ∈ {1, 2}, by (C/J i )⊗ min A i to obtain an immediate generalization of this noncommutative join construction. In the classical setting (i.e. for C commutative), this generalization would correspond to replacing the interval [0, 1] with its endpoints 0 and 1 by a compact Hausdorff space with two disjoint closed subsets.

Join of a principal H-comodule algebra with H
Since a diagonal coaction is not in general an algebra homomorphism, to obtain an equivariant version of our noncommutative join construction, we need to modify Definition 3.1 in the spirit of (1.5)-(1.6). We also need to change the setting from C*-algebraic to algebraic to avoid analytical complications that are tackled in [3].
Definition 4.1. Let ∆ P : P → P ⊗ H be a right coaction making P a comodule algebra. We call the algebra the join algebra of P and H. Here ev r : C([0, 1]) f → f (r) ∈ C is the evaluation map at r. Proof. Note first that Then, the above tensor belongs to C ⊗ H ⊗ H for r = 0 and to (4.18) (id ⊗ ∆)(∆ P (P )) = (∆ P ⊗ id)(∆ P (P )) ⊆ ∆ P (P ) ⊗ H for r = 1.
The main result of this paper is: If P is a principal right H-comodule algebra, then the join right H-comodule algebra P * H is principal.
Proof. Let : H→P ⊗ P be a strong connection on P , and let t : [0, 1] → C be the inclusion map. Then the bicolinearity of implies that the linear map corestricts to (P * H) ⊗ (P * H). The bicolinearity of the thus corestricted˜ is evident. Finally, taking advantage of the right colinearity and the splitting property of , we check that Now the claim follows from Lemma 2.3.
Example 4.4. In particular, taking P = H to be the Hopf algebra of SU q (2), we can conclude the principality of Pflaum's noncommutative instanton bundle [7]. The fact that this bundle is not trivial follows from [4]. In the classical setting, except for the trivial group, all joins G * G, where G is a compact Hausdorff group, are non-trivial as principal G-bundles. We conjecture the same is true for all compact quantum groups [10].

Piecewise structure
Let P be a principal right H-comodule algebra. First we define the following algebras The P i 's become H-comodule algebras for the coactions obtained by the restrictions and corestrictions of id ⊗ id ⊗ ∆, and the subalgebras of H-coaction invariants are respectively Now one can identify P with the pullback comodule algebra (5.22) {(p, q) ∈ P 1 ⊕ P 2 | (ev 1 ⊗ id)(p) = (ev 0 ⊗ id)(q)} of the P i 's along the right H-colinear algebra homomorphisms (5.23) (ev 1 ⊗ id) : P 1 −→ P ⊗ H, (ev 0 ⊗ id) : P 2 −→ P ⊗ H.