Shifts of finite type, CuntzKrieger algebras and their algebraic - - PowerPoint PPT Presentation

Shifts of finite type, Cuntz–Krieger algebras and their algebraic analogues, groupoids, and inverse semigroups

T

• ke Meier Carlsen

Facets of Irreversibility: Inverse Semigroups, Groupoids, and Operator Algebras University of Oslo, 4–8 December 2017

Theorem 1

Let A and B be finite square {0, 1}-matrices with no zero rows and no zero columns, and let R be an indecomposable reduced commutative ring with unit. The following are equivalent.

1 The one-sided shift spaces (XA, σA) and (XB, σB) are

continuously orbit equivalent.

2 The groupoids GA and GB are isomorphic as topological

groupoids.

3 The inverse semigroups SA and SB are isomorphic. 4 The Cuntz–Krieger algebras OA and OB are isomorphic by

a diagonal-preserving isomorphism.

5 The Steinberg algebras RGA and RGB are isomorphic by a

diagonal-preserving isomorphism.

T .M. Carlsen SFTs, Cuntz–Krieger and Steinberg algebras, groupoids, and inv. semigroups Page 1/5

Theorem 2

Let A and B be finite square {0, 1}-matrices with no zero rows and no zero columns, and let R be an indecomposable commutative ring with unit. The following are equivalent.

1 The one-sided shift spaces (XA, σA) and (XB, σB) are

eventually conjugate.

2 There is an isomorphism ϕ : GA → GB such that cA = cB ◦ ϕ. 3 There is an isomorphism ψ : Se A → Se B such that eA = eB ◦ ψ. 4 The Cuntz–Krieger algebras OA and OB are isomorphic by

a diagonal-preserving isomorphism that intertwines the gauge actions λA

t and λB t . 5 The Steinberg algebras RGA and RGB are isomorphic by a

graded diagonal-preserving isomorphism.

T .M. Carlsen SFTs, Cuntz–Krieger and Steinberg algebras, groupoids, and inv. semigroups Page 2/5

Theorem 3

Let A and B be finite square {0, 1}-matrices with no zero rows and no zero columns, and let R be an indecomposable commutative ring with unit. The following are equivalent.

1 The one-sided shift spaces (XA, σA) and (XB, σB) are

conjugate.

2 There is an isomorphism ϕ : GA → GB such that

ϕ ◦ εA = εB ◦ ϕ.

3 There is an isomorphism ψ : SA → SB such that ψ(Sr A) = Sr B

and ψ ◦ rA = rB ◦ ψ.

4 The Cuntz–Krieger algebras OA and OB are isomorphic by

a diagonal-preserving isomorphism that intertwines the positive maps τA and τB.

5 The Steinberg algebras RGA and RGB are isomorphic by a

diagonal-preserving isomorphism that intertwines κA and κB.

T .M. Carlsen SFTs, Cuntz–Krieger and Steinberg algebras, groupoids, and inv. semigroups Page 3/5

Theorem 4

Let A and B be finite square {0, 1}-matrices with no zero rows and no zero columns, and let R be an indecomposable reduced commutative ring with unit. The following are equivalent.

1 The two-sided shift spaces (XA, σA) and (XB, σB) are

flow-equivalent.

2 The groupoids GA × R and GB × R are isomorphic. 3 The inverse semigroups ˜

SA and ˜ SB are isomorphic.

4 The stabilised Cuntz–Krieger algebras OA ⊗ K and OB ⊗ K

are isomorphic by a diagonal-preserving isomorphism.

5 The algebras RGA ⊗ M∞(R) and RGB ⊗ M∞(R) are

isomorphic by a diagonal-preserving isomorphism.

T .M. Carlsen SFTs, Cuntz–Krieger and Steinberg algebras, groupoids, and inv. semigroups Page 4/5

Theorem 5

Let A and B be finite square {0, 1}-matrices with no zero rows and no zero columns, and let R be an indecomposable commutative ring with unit. The following are equivalent.

1 The two-sided shift spaces (XA, σA) and (XB, σB) are

conjugate.

2 There is an isomorphism ϕ : GA × R → GB × R such that

˜ cA = ˜ cB ◦ ϕ.

3 There is an isomorphism ψ : ˜

Se

A → ˜

Se

B such that ˜

eA = ˜ eB ◦ ψ.

4 The stabilised Cuntz–Krieger algebras OA ⊗ K and OB ⊗ K

are isomorphic by a diagonal-preserving isomorphism that intertwines the actions λA

t ⊗ idK and λB t ⊗ idK. 5 The algebras RGA ⊗ M∞(R) and RGB ⊗ M∞(R) are

isomorphic by a graded diagonal-preserving isomorphism.

T .M. Carlsen SFTs, Cuntz–Krieger and Steinberg algebras, groupoids, and inv. semigroups Page 5/5