The Zappa-Sz ep product of a Fell bundle by a groupoid Boyu Li - - PowerPoint PPT Presentation

The Zappa-Sz´ ep product of a Fell bundle by a groupoid

Boyu Li

University of Victoria

November 4th, 2020 Joint work with Anna Duwenig

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Let G, H be two groups. Recall the semi-direct product encodes an H-action on G: (h, x) ∈ H × G → h · x ∈ G.

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Let G, H be two groups. Recall the semi-direct product encodes an H-action on G: (h, x) ∈ H × G → h · x ∈ G. The semi-direct product group is defined as: G ⋊ H = {(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).

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Let G, H be two groups. Now, the Zappa-Sz´ ep product of G and H encodes an additional “G-action on H” (with some compatibility conditions with the H-action on G): (h, x) ∈ H × G → h|x ∈ H.

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Let G, H be two groups. Now, the Zappa-Sz´ ep product of G and H encodes an additional “G-action on H” (with some compatibility conditions with the H-action on G): (h, x) ∈ H × G → h|x ∈ H. The (external) Zappa-Sz´ ep product group is defined as: G ⊲ ⊳ H = {(x, h) : x ∈ G, h ∈ H}, with multiplication and inverse: (x, h)(y, k) = (x(h · y), h|yk), (x, h)−1 = (h−1 · x, h−1|x−1).

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Let G, H be two groups. Now, the Zappa-Sz´ ep product of G and H encodes an additional “G-action on H” (with some compatibility conditions with the H-action on G): (h, x) ∈ H × G → h|x ∈ H. The (external) Zappa-Sz´ ep product group is defined as: G ⊲ ⊳ H = {(x, h) : x ∈ G, h ∈ H}, with multiplication and inverse: (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 all h ∈ H, x ∈ G), this coincides with the semi-direct product.

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Let G, H be two ´ etale groupoids.

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Let G, H be two ´ etale groupoids. We say they are matching if G(0) = H(0) and there exists continuous H-action and G-restriction maps: (h, x) → h · x ∈ G, s(h) = r(x), (h, x) → h|x ∈ H, s(h) = r(x),

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Let G, H be two ´ etale groupoids. We say they are matching if G(0) = H(0) and there exists continuous H-action and G-restriction maps: (h, x) → h · x ∈ G, s(h) = r(x), (h, x) → 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)

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

The (external) Zappa-Sz´ ep product groupoid is defined as G ⊲ ⊳ H = {(x, h) : x ∈ G, h ∈ H, r(h) = s(x)}, with multiplication and inverse: (x, h)(y, g) = (x(h · y), h|yg), (x, h)−1 = (h−1 · x−1, h−1|x−1).

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

The (external) Zappa-Sz´ ep product groupoid is defined as G ⊲ ⊳ H = {(x, h) : x ∈ G, h ∈ H, r(h) = s(x)}, with multiplication and inverse: (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 ´ etale if and only if G and H are both ´ etale .

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Now in the realm of operator algebra, semi-direct product is related to the 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: A ⋊alg

α 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).

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Now in the realm of operator algebra, semi-direct product is related to the 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: A ⋊alg

α 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-Sz´ ep analogue of this?

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Question

What is a Zappa-Sz´ ep analogue of this?

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Question

What is a Zappa-Sz´ ep analogue of this? Several recent studies on very specific examples of Zappa-Sz´ ep type

• perator algebras:

C∗-algebra of self-similar groups C∗-algebra of self-similar graphs and k-graphs Groupoid C∗-algebra of the Zappa-Sz´ ep groupoids

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Background and Motivation

Question

What is a Zappa-Sz´ ep analogue of this? Several recent studies on very specific examples of Zappa-Sz´ ep type

• perator algebras:

C∗-algebra of self-similar groups C∗-algebra of self-similar graphs and k-graphs Groupoid C∗-algebra of the Zappa-Sz´ ep 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-trivial

• way. 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-Sz´ ep structure.

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Fell Bundle

A Fell bundle provides a grading:

Definition

A Fell bundle B = (B, p) over a groupoid G is a upper semicontinuous Banach bundle equipped with continuous multiplication and involution such that For each (x, y) ∈ G(2), Bx · By ⊂ Bxy. The multiplication is bilinear and associative. For any b, c ∈ B, b · c ≤ bc. 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 = b2 = b∗2. For any b ∈ B, b∗b ≥ 0 as an element in Bs(p(b)).

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Compatible Action

We now need a compatible action:

Definition

Let B = (B, p) be a Fell bundle over an ´ etale groupoid G, and let H be a matching ´ etale groupoid. A (G, H)-compatible H-action on B is a continuous map: β : (h, b) → 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∗).

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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 of C∗-algebras.

Proposition

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

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Zappa-Sz´ ep Product of a Fell bundle by a groupoid

Definition

Let B = (B, p) be a Fell bundle over an ´ etale groupoid G and let H be a matching ´ etale 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).

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Zappa-Sz´ ep Product of a Fell bundle by a groupoid

Theorem (Duwenig, L.)

The Banach bundle C = (C, q) is a Fell bundle over G ⊲ ⊳ H. We call it the Zappa-Sz´ ep product of B by H, denoted by B ⊲ ⊳β H.

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Zappa-Sz´ ep Product of a Fell bundle by a groupoid

Theorem (Duwenig, L.)

The Banach bundle C = (C, q) is a Fell bundle over G ⊲ ⊳ H. We call it the Zappa-Sz´ ep product of B by H, denoted by B ⊲ ⊳β H.

Theorem (Internal Zappa-Sz´ ep Product)

Suppose K is a groupoid and G, H are subgroupoids. If every k ∈ K can be uniquely written as k = gh for some g ∈ G and h ∈ H. Then K is isomorphic to a Zappa-Sz´ ep product group G ⊲ ⊳ H.

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Zappa-Sz´ ep Product of a Fell bundle by a groupoid

Theorem (Duwenig, L.)

The Banach bundle C = (C, q) is a Fell bundle over G ⊲ ⊳ H. We call it the Zappa-Sz´ ep product of B by H, denoted by B ⊲ ⊳β H.

Theorem (Internal Zappa-Sz´ ep Product)

Suppose K is a groupoid and G, H are subgroupoids. If every k ∈ K can be uniquely written as k = gh for some g ∈ G and h ∈ H. Then K is isomorphic to a Zappa-Sz´ ep product group G ⊲ ⊳ H.

Theorem (Duwenig, L.)

Suppose C = (C, q) is a Fell bundle over an ´ etale groupoid K, and G, H are 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 = buh

for some b ∈ j∗

G(C) and h ∈ H. Then C is isomorphic to B ⊲

⊳β H, where B = j∗

G(C) and βh(b) = uhbu∗ h|p(b).

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Examples

For an ´ etale groupoid G, define its groupoid Fell bundle to be B(G) := C × G. For a matching pair of ´ etale groupoids G, H, define an action 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).

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Examples

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

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

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Examples

Define H = G(0) ⋊β G be the “transformation groupoid”: define multiplication (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.

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Representations and C∗-algebra

Given a Fell bundle B over an ´ etale groupoid G, define Γc(G, B) = {σ : G → B|σ is continuous, cpt-supp, σ(x) ∈ Bx}. This is a ∗-algebra by multiplication: στ(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 σI,r = sup

v∈G(0)(

• r(x)=v

σ(x)), σI,s = sup

v∈G(0)(

• s(x)=v

σ(x)).

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Representations and C∗-algebra

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 of the Fell bundle B, denoted by C∗(B). Every non-degenerate I-norm decreasing ∗-representation corresponds to a “strict”-representation (µ, G(0) ∗ H, π), and vice versa.

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Representations and C∗-algebra

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

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Representations and C∗-algebra

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

Definition

A covariant representation for B ⊲ ⊳β H consists of a “strict representation” (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).

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Representations and C∗-algebra

Theorem (Duwenig, L.)

Every covariant representation of B ⊲ ⊳β H “integrates” into an I-norm decreasing ∗-representation of Γc(G, B).

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Representations and C∗-algebra

Theorem (Duwenig, L.)

Every covariant representation of B ⊲ ⊳β H “integrates” into an I-norm decreasing ∗-representation of Γc(G, B). Conversely, assuming that Bu is unital for all u ∈ G(0), every non-degenerate I-norm decreasing ∗-representation “dis-integrate” as a covariant representation.

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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 dense range.

Theorem (BPRRW 2017)

The groupoid C∗-algebra C∗(G ⊲ ⊳ H) of ´ etale groupoids G, H is a C∗-blend of C∗(G) and C∗(H).

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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 dense range.

Theorem (BPRRW 2017)

The groupoid C∗-algebra C∗(G ⊲ ⊳ H) of ´ etale groupoids G, H is a C∗-blend of C∗(G) and C∗(H).

Theorem (Duwenig, L.)

Assuming that Bu is unital for all u ∈ G(0), C∗(B ⊲ ⊳β H) is a C∗-blend

• f C∗(B) and C∗(H).

Boyu Li (University of Victoria) Zappa-Sz´ ep product of Fell bundle

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Regular representation

Recall in a C∗-dynamical system (A, G, α), any representation ρ : A → B(H) can be “lifted” to a covariant representation (˜ ρ, U) on B(H ⊗ ℓ2(G)) by ˜ ρ(a)ξ ⊗ eg = ρ(αg−1(a))ξ ⊗ eg; Uhξ ⊗ eg = ξ ⊗ ehg.

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Regular representation

Recall in a C∗-dynamical system (A, G, α), any representation ρ : A → B(H) can be “lifted” to a covariant representation (˜ ρ, U) on B(H ⊗ ℓ2(G)) by ˜ ρ(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 a discrete group G. Then we can also “lift” it to a covariant representation (˜ π, U) of B ⊲ ⊳β H on B(H ⊗ ℓ2(G ⊲ ⊳ H)), by: ˜ π(b)ξ ⊗ e(x,h) = π(βh−1|(p(b)x)−1(b))ξ ⊗ e(π(b)x,h), and Ukξ ⊗ 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).

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Thank you

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