The Zappa-Sz ep product of a Fell bundle by a groupoidquigg/seminar/20f/slides/201104.pdfThe Zappa-Sz ep product of a Fell bundle by a groupoid Boyu Li University of Victoria November

The Zappa-Szép product of a Fell bundle by agroupoid

Boyu Li

University of Victoria

November 4th, 2020

Joint work with Anna Duwenig

Boyu Li (University of Victoria) Zappa-Szép product of Fell bundle Nov. 4th, 2020 1 / 22

Background and Motivation

Let G,H be two groups. Recall the semi-direct product encodes anH-action on G:

(h, x) ∈ H ×G 7→ h · x ∈ G.

The semi-direct product group is defined as:

GoH = {(x, h) : x ∈ G, h ∈ H},

with multiplication and inverse:

(x, h)(y, k) = (x(h · y), hk), (x, h)−1 = (h−1 · x, h−1).

Let G,H be two groups. Recall the semi-direct product encodes anH-action on G:

(h, x) ∈ H ×G 7→ h · x ∈ G.

The semi-direct product group is defined as:

GoH = {(x, h) : x ∈ G, h ∈ H},

(x, h)(y, k) = (x(h · y), hk), (x, h)−1 = (h−1 · x, h−1).

Let G,H be two groups. Now, the Zappa-Szép product of G and Hencodes an additional “G-action on H” (with some compatibilityconditions with the H-action on G):

(h, x) ∈ H ×G 7→ h|x ∈ H.

The (external) Zappa-Szép product group is defined as:

G ./ H = {(x, h) : x ∈ G, h ∈ H},

(x, h)(y, k) = (x(h · y), h|yk), (x, h)−1 = (h−1 · x, h−1|x−1).

Note that when the G-restriction map is trivial (that is h|x = h for allh ∈ H,x ∈ G), this coincides with the semi-direct product.

(h, x) ∈ H ×G 7→ h|x ∈ H.

G ./ H = {(x, h) : x ∈ G, h ∈ H},

(h, x) ∈ H ×G 7→ h|x ∈ H.

G ./ H = {(x, h) : x ∈ G, h ∈ H},

Let G,H be two étale groupoids.

We say they are matching ifG(0) = H(0) and there exists continuous H-action and G-restrictionmaps:

(h, x) 7→ h · x ∈ G, s(h) = r(x),(h, x) 7→ h|x ∈ H, s(h) = r(x),

such that:

(ZS1) (h1h2) · x = h1 · (h2 · x) (ZS2) h|xy = (h|x)|y(ZS3) rG(x) · x = x (ZS4) h|sH(h) = h(ZS5) rG(h · x) = rH(h) (ZS6) sH(h|x) = sG(x)(ZS7) h · (xy) = (h · x)(h|x · y) (ZS8) (hk)|x = h|k·xk|x(ZS9) sG(h · x) = rH(h|x)

Let G,H be two étale groupoids. We say they are matching ifG(0) = H(0) and there exists continuous H-action and G-restrictionmaps:

such that:

Let G,H be two étale groupoids. We say they are matching ifG(0) = H(0) and there exists continuous H-action and G-restrictionmaps:

such that:

The (external) Zappa-Szép product groupoid is defined as

G ./ H = {(x, h) : x ∈ G, h ∈ H, r(h) = s(x)},

(x, h)(y, g) = (x(h · y), h|yg), (x, h)−1 = (h−1 · x−1, h−1|x−1).

Theorem (Brownlowe, Pask, Ramagge, Robertson, Whittaker,2017)

If G,H are mathcing groupoids, then G ./ H is étale if and only if Gand H are both étale .

The (external) Zappa-Szép product groupoid is defined as

G ./ H = {(x, h) : x ∈ G, h ∈ H, r(h) = s(x)},

(x, h)(y, g) = (x(h · y), h|yg), (x, h)−1 = (h−1 · x−1, h−1|x−1).

Theorem (Brownlowe, Pask, Ramagge, Robertson, Whittaker,2017)

If G,H are mathcing groupoids, then G ./ H is étale if and only if Gand H are both étale .

Now in the realm of operator algebra, semi-direct product is related tothe crossed product:In its simplest form, let (A, H, α) be a C∗-dynamical system. The(discrete) group H act on a C∗-algebra A by a ∗-automorphic action α.One may form the algebraic crossed product:

Aoalgα H := {(a, g) : a ∈ A, g ∈ H}.

We can put a ∗-algebra structure by

(a, g)(b, h) = (aαg(b), gh), (a, g)∗ = (αg−1(a

∗), g−1).

Question

What is a Zappa-Szép analogue of this?

Now in the realm of operator algebra, semi-direct product is related tothe crossed product:In its simplest form, let (A, H, α) be a C∗-dynamical system. The(discrete) group H act on a C∗-algebra A by a ∗-automorphic action α.One may form the algebraic crossed product:

Aoalgα H := {(a, g) : a ∈ A, g ∈ H}.

We can put a ∗-algebra structure by

(a, g)(b, h) = (aαg(b), gh), (a, g)∗ = (αg−1(a

∗), g−1).

Question

Several recent studies on very specific examples of Zappa-Szép typeoperator algebras:

C∗-algebra of self-similar groups

C∗-algebra of self-similar graphs and k-graphs

Groupoid C∗-algebra of the Zappa-Szép groupoids

To build a general framework, there are two key ingredients:

The C∗-algebra A has to “act” on the group H in a non-trivialway. This forces some kind of grading on A.We also need to define a notion of the ∗-automorphic action α,that is compatible with the Zappa-Szép structure.

Question

Fell Bundle

A Fell bundle provides a grading:

Definition

A Fell bundle B = (B, p) over a groupoid G is a upper semicontinuousBanach bundle equipped with continuous multiplication and involutionsuch that

For each (x, y) ∈ G(2), Bx · By ⊂ Bxy.The multiplication is bilinear and associative.

For any b, c ∈ B, ‖b · c‖ ≤ ‖b‖‖c‖.For any x ∈ G, B∗x ⊂ Bx−1.The involution is conjugate linear.

For any b, c ∈ B, (bc)∗ = c∗b∗ and b∗∗ = b.For any b ∈ B, ‖b∗b‖ = ‖b‖2 = ‖b∗‖2.For any b ∈ B, b∗b ≥ 0 as an element in Bs(p(b)).

Compatible Action

We now need a compatible action:

Definition

Let B = (B, p) be a Fell bundle over an étale groupoid G, and let H bea matching étale groupoid. A (G,H)-compatible H-action on B is acontinuous map:

β : (h, b) 7→ βh(b), s(h) = r(p(b)),

satisfying:

βh is a linear map from Bx to Bh·x for all s(h) = r(x).For any (g, h) ∈ H(2), βg ◦ βh = βgh.For any u ∈ H(0), βu is the identity map.For any bc ∈ B and r(p(b)) = s(h), βh(bc) = βh(b)βh|p(b)(c).For any b ∈ B with r(p(b)) = s(h), βh(b)∗ = βh|p(b)(b

∗).

Compatible Action

The maps {βh} are not ∗-automorphic as in the semi-crossed product.However, they do enjoy some nice properties:

Proposition

For each h ∈ H, βh : Bs(h) → Br(h) is an injective ∗-isomorphism ofC∗-algebras.

Proposition

For each h ∈ H and x ∈ G with s(h) = r(x), βh : Bx → Bh·x isisometric.

Zappa-Szép Product of a Fell bundle by a groupoid

Definition

Let B = (B, p) be a Fell bundle over an étale groupoid G and let H be amatching étale groupoid. Let β be a (G,H)-compatible H-action on B.Define a Banach bundle C = (C, q) by

C = {(b, h) : b ∈ B, h ∈ H, s(p(b)) = r(h)},

and q(b, h) = (p(b), h) ∈ G ./ H. Define multiplication by:

(b, h)(c, k) = (bβh(c), h|p(c)k), s(h) = r(p(c)),

and involution by:

(b, h)∗ = (βh−1(b∗), h−1|p(b)−1).

Theorem (Duwenig, L.)

The Banach bundle C = (C, q) is a Fell bundle over G ./ H. We call itthe Zappa-Szép product of B by H, denoted by B ./β H.

Theorem (Internal Zappa-Szép Product)

Suppose K is a groupoid and G,H are subgroupoids. If every k ∈ K canbe uniquely written as k = gh for some g ∈ G and h ∈ H. Then K isisomorphic to a Zappa-Szép product group G ./ H.

Suppose C = (C, q) is a Fell bundle over an étale groupoid K, and G,Hare subgroupoids of K. If there exists a “continuous unitary section”u : H → j∗H(C), and every element c ∈ C is a unique product of c = buhfor some b ∈ j∗G(C) and h ∈ H. Then C is isomorphic to B ./β H, whereB = j∗G(C) and βh(b) = uhbu∗h|p(b).

Examples

For an étale groupoid G, define its groupoid Fell bundle to beB(G) := C× G. For a matching pair of étale groupoids G,H, define anaction β on B(G) by

βh(z, x) = (z, h · x) if s(h) = r(x).

Then β is an (G,H)-compatible action on B(G). We can verify that

B(G) ./β H = B(G ./ H).

Examples

Example (Kaliszewski, Muhly, Quigg and Williams, 2009)

Let G be a group acting on a groupoid G by β : G→ Aut(G). Nowsuppose B is a Fell bundle over G. Let α : G→ Aut(B) be an action ofG on B with an associated action t · x of G on G, such thatp(αt(b)) = t · p(b). Then, define a Banach bundle B ×α G byq(b, t) = (p(b), t), and multiplication

(bx, t)(cy, s) = (bxαt(cy), ts), s(x) = r(t · y),

and involution(bx, t)

∗ = (αt−1(b∗x), t

−1).

Then B ×α G is a Fell bundle.

Examples

Define H = G(0) oβ G be the “transformation groupoid”: definemultiplication

(u, t)(v, s) = (u, ts), if v = βt−1(u);

and inverse(u, t)−1 = (βt−1(u), t

−1).

Then G,H becomes a matching pair of groupoids with H-action map(u, t) · x = t · x and (trivial) G-restriction map (u, t)|x = (u, t).The map α corresponds to a (G,H)-compatible H-action β by

β(u,t)(b) = αt(b).

We have that B ×α G is isomorphic to B ./β G.

Representations and C∗-algebra

Given a Fell bundle B over an étale groupoid G, define

Γc(G,B) = {σ : G → B|σ is continuous, cpt-supp, σ(x) ∈ Bx}.

This is a ∗-algebra by multiplication:

σ�τ(x) =∑

r(y)=r(x)

σ(y)τ(y−1x),

and involution:σ∗(x) = σ(x−1)∗.

Define the I-norm to be the max of

‖σ‖I,r = supv∈G(0)

(∑r(x)=v

‖σ(x)‖), ‖σ‖I,s = supv∈G(0)

(∑s(x)=v

‖σ(x)‖).

Define the universal norm by

‖σ‖∞ = sup{‖L(σ)‖ : L is an I-norm decreasing ∗ -representation}.

The closure of Γc(G,B) under this norm is the universal C∗-algebra ofthe Fell bundle B, denoted by C∗(B).Every non-degenerate I-norm decreasing ∗-representation correspondsto a “strict”-representation (µ,G(0) ∗H, π), and vice versa.

Recall in the case of the C∗-crossed product: representation ofΓc(G,A) is closely related to covariant representation (π, U) whereUgπ(a) = π(αg(a))Ug.

Definition

A covariant representation for B ./β H consists of a “strictrepresentation” (µ,G(0) ∗H, π) and a “unitary representation” U of H,such that for any h ∈ H and b ∈ B with s(h) = r(p(b)),

Uhπ(b) = π(βh(b))Uh|p(b) .

Recall in the case of the C∗-crossed product: representation ofΓc(G,A) is closely related to covariant representation (π, U) whereUgπ(a) = π(αg(a))Ug.

Definition

A covariant representation for B ./β H consists of a “strictrepresentation” (µ,G(0) ∗H, π) and a “unitary representation” U of H,such that for any h ∈ H and b ∈ B with s(h) = r(p(b)),

Uhπ(b) = π(βh(b))Uh|p(b) .

Every covariant representation of B ./β H “integrates” into an I-normdecreasing ∗-representation of Γc(G,B).

Conversely, assuming that Bu is unital for all u ∈ G(0), everynon-degenerate I-norm decreasing ∗-representation “dis-integrate” as acovariant representation.

Every covariant representation of B ./β H “integrates” into an I-normdecreasing ∗-representation of Γc(G,B).

Conversely, assuming that Bu is unital for all u ∈ G(0), everynon-degenerate I-norm decreasing ∗-representation “dis-integrate” as acovariant representation.

C∗-blend

Definition

A C∗-blend is a quintuple (A1,A2, i1, i2,A) where ik : Ak → A is a∗-homomorphism for k = 1, 2, and i1 ⊗ i2 : A1 ⊗C A2 → A has denserange.

Theorem (BPRRW 2017)

The groupoid C∗-algebra C∗(G ./ H) of étale groupoids G,H is aC∗-blend of C∗(G) and C∗(H).

Assuming that Bu is unital for all u ∈ G(0), C∗(B ./β H) is a C∗-blendof C∗(B) and C∗(H).

C∗-blend

Definition

A C∗-blend is a quintuple (A1,A2, i1, i2,A) where ik : Ak → A is a∗-homomorphism for k = 1, 2, and i1 ⊗ i2 : A1 ⊗C A2 → A has denserange.

Theorem (BPRRW 2017)

The groupoid C∗-algebra C∗(G ./ H) of étale groupoids G,H is aC∗-blend of C∗(G) and C∗(H).

Assuming that Bu is unital for all u ∈ G(0), C∗(B ./β H) is a C∗-blendof C∗(B) and C∗(H).

Regular representation

Recall in a C∗-dynamical system (A, G, α), any representationρ : A → B(H) can be “lifted” to a covariant representation (ρ̃, U) onB(H⊗ `2(G)) by

ρ̃(a)ξ ⊗ eg = ρ(αg−1(a))ξ ⊗ eg;Uhξ ⊗ eg = ξ ⊗ ehg.

Now let π : B → B(H) be a representation of a Fell bundle B over adiscrete group G. Then we can also “lift” it to a covariantrepresentation (π̃, U) of B ./β H on B(H⊗ `2(G ./ H)), by:

π̃(b)ξ ⊗ e(x,h) = π(βh−1|(p(b)x)−1 (b))ξ ⊗ e(π(b)x,h),

andUkξ ⊗ e(x,h) = ξ ⊗ e(1,k)(x,h) = ξ ⊗ e(k·x,k|xh).

Corollary

When G and H are discrete groups, there is a natural injective∗-homomorphism i : C∗(B)→ C∗(B ./β H).

Regular representation

Recall in a C∗-dynamical system (A, G, α), any representationρ : A → B(H) can be “lifted” to a covariant representation (ρ̃, U) onB(H⊗ `2(G)) by

ρ̃(a)ξ ⊗ eg = ρ(αg−1(a))ξ ⊗ eg;Uhξ ⊗ eg = ξ ⊗ ehg.

Now let π : B → B(H) be a representation of a Fell bundle B over adiscrete group G. Then we can also “lift” it to a covariantrepresentation (π̃, U) of B ./β H on B(H⊗ `2(G ./ H)), by:

π̃(b)ξ ⊗ e(x,h) = π(βh−1|(p(b)x)−1 (b))ξ ⊗ e(π(b)x,h),

andUkξ ⊗ e(x,h) = ξ ⊗ e(1,k)(x,h) = ξ ⊗ e(k·x,k|xh).

Corollary

When G and H are discrete groups, there is a natural injective∗-homomorphism i : C∗(B)→ C∗(B ./β H).

Thank you