Arens-Michael envelopes of Ore extensions · Peter Kosenko (Higher School of Economics) Arens-Michael envelopes of Ore extensions July 11, 2017 5 / 18 The Arens-Michael envelope is

Arens-Michael envelopes of Ore extensions

Peter Kosenko

Higher School of Economics

July 11, 2017

Peter Kosenko (Higher School of Economics) Arens-Michael envelopes of Ore extensions July 11, 2017 1 / 18

Arens-Michael algebras

All algebras are assumed to be complex, unital and associative.

DefinitionAn Arens-Michael algebra is a complete locally convex algebra with topology thatcan be generated by a family of submultiplicative seminorms.

Here are some examples:1 Any Banach algebra is an Arens-Michael algebra.2 For a locally compact topological space X the algebra of continuous

functions C(X), endowed with the compact-open topology, is anArens-Michael algebra.

3 For any open U ⊂ Cn the algebra of holomorphic functions O(U), endowedwith the compact-open topology, is an Arens-Michael algebra.


Arens-Michael envelopes

DefinitionLet A be an algebra. The Arens-Michael envelope of A is a pair (A, iA), where Ais an Arens-Michael algebra and iA : A→ A is an algebra homomorphism,satisfying the following universal property: for any Arens-Michael algebra B andalgebra homomorphism ϕ : A→ B there exists a unique continuous algebrahomomorphism ϕ : A→ B such that the following diagram commutes:

A B

A

ϕ

iA ϕ

The Arens-Michael envelope of any algebra exists and unique. It is isomorphic tothe completion of A with respect to the family of all submultiplicative seminormson A. Also notice that due to the Arens-Michael decomposition it suffices tocheck the universal property when B is a Banach algebra.


ExamplesTheorem (J. Taylor, 1972)The Arens-Michael envelope of C[t1, . . . tn] is isomorphic to O(Cn).

Proposition (J. Taylor, 1972)Let Fn be the free algebra on n non-commuting variables xi. The Arens-Michaelenvelope of Fn can be described as follows:

Fn ' {f =∑w∈Wn

awxw : ||f ||ρ =

∑w∈Wn

|aw|ρ|w| <∞, 0 < ρ <∞},

where Wn =⊔∞d=0{1, . . . , n}d and xw = xw1xw2 . . . xwn

. The resultingArens-Michael algebra will be denoted by Fn.

Theorem (J. Taylor, 1972)Let g be a finite-dimensional semisimple Lie algebra. The Arens-Michael envelopeof U(g) is isomorphic to the direct product

∏V ∈g Mat(V ), where g is the set of

finite dimensional irreducible reps of g (or the equivalence classes, to be exact).




Fn ' {f =∑w∈Wn

awxw : ||f ||ρ =

∑w∈Wn

|aw|ρ|w| <∞, 0 < ρ <∞},









Fn ' {f =∑w∈Wn

awxw : ||f ||ρ =

∑w∈Wn

|aw|ρ|w| <∞, 0 < ρ <∞},







The Arens-Michael envelope is a left adjoint functor to the forgetful functor,therefore it commutes with quotients. In other words, we have the followingtheorem:

TheoremLet A be an algebra and suppose that I ⊂ A is a two-sided ideal. Denote by Jthe closure of iA(I) in A. Then J is a two-sided ideal in A and the inducedhomomorphism A/I → A/J extends to a topological algebra isomorphism

(A/I) ' (A/J )

As a corollary (not quite trivial), we get the following result:

TheoremFor any complex affine algebraic variety X we have

(Oreg(X)) ' Ohol(X).


The Arens-Michael envelope is a left adjoint functor to the forgetful functor,therefore it commutes with quotients. In other words, we have the followingtheorem:

TheoremLet A be an algebra and suppose that I ⊂ A is a two-sided ideal. Denote by Jthe closure of iA(I) in A. Then J is a two-sided ideal in A and the inducedhomomorphism A/I → A/J extends to a topological algebra isomorphism

(A/I) ' (A/J )

As a corollary (not quite trivial), we get the following result:

TheoremFor any complex affine algebraic variety X we have

(Oreg(X)) ' Ohol(X).


It turns out that iA : A→ A isn’t always injective!

PropositionLet A = C 〈x, y〉 /([x, y]− 1). Then A ' 0.

The proof relies on the fact that if x, y ∈ B, where B is a non-zero Banachalgebra, then [x, y] 6= 1.As a matter of fact, let us formulate the following proposition:

PropositionConsider an element f ∈ Fn. Then the following statements are equivalent:

1 there exist a Banach algebra B and elements x1, . . . , xn ∈ B such thatf(x) = 0.

2 if I ⊂ Fn is the smallest two-sided ideal which contains f , then

(Fn/I) 6= 0.

Such elements are called admissible. For example, xy − yx− 1 is not admissible.




The proof relies on the fact that if x, y ∈ B, where B is a non-zero Banachalgebra, then [x, y] 6= 1.

As a matter of fact, let us formulate the following proposition:




(Fn/I) 6= 0.









(Fn/I) 6= 0.









(Fn/I) 6= 0.



Ore extensionsDefinitionLet A be an algebra. For any endomorphism α : A→ A denote by Aα a anA-bimodule, which coincides with A as a left A-module, and whose structure ofright A-module is defined by x ◦ a = xα(a), where x ∈ Aα, a ∈ A.In a similar fashion one defines αA.

Let us recall the definition of Ore extensions:DefinitionLet A be an algebra, α ∈ End(A) and δ ∈ Der(A, αA), in other words,

δ(ab) = α(a)δ(b) + δ(a)b.

Then the Ore extension of A wrt α and δ is the vector spaceA[t;α, δ] =

{∑ni=0 ait

i : ai ∈ A}

with the multiplication, which is uniquelydetermined by

ta = α(a)t+ δ(a) (a ∈ A).

Also, if δ = 0 and α is invertible, then one can define the Laurent Ore extensionA[t, t−1;α] =

{∑ni=−n ait

i : ai ∈ A}

with the multiplication defined similarly.






{∑ni=0 ait

i : ai ∈ A}


ta = α(a)t+ δ(a) (a ∈ A).


{∑ni=−n ait

i : ai ∈ A}

with the multiplication defined similarly.






{∑ni=0 ait

i : ai ∈ A}


ta = α(a)t+ δ(a) (a ∈ A).


{∑ni=−n ait

i : ai ∈ A}

with the multiplication defined similarly.Peter Kosenko (Higher School of Economics) Arens-Michael envelopes of Ore extensions July 11, 2017 7 / 18

DefinitionLet A is an Arens-Michael algebra. A family of continuous linear maps T ⊂ L(E)is called m-localizable ⇐⇒ there exists a generating family of submultiplicativeseminorms {|| · ||ν : ν ∈ Λ} on A such that for every T ∈ T and ν ∈ Λ thereexists C > 0 such that ||T (x)||ν ≤ C||x||ν for every x ∈ A.

Theorem (A. Pirkovskii)Let A be an algebra, α ∈ End(A) and δ ∈ Der(A, αA), such that (α, δ) is anm-localizable pair of morphisms A→ A. Fix a generating family of seminorms{‖·‖λ : λ ∈ Λ} on A. Then the locally convex space

O(C, A) =

{f =

∞∑k=0

akzk :

∞∑k=0

‖ak‖λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

}

1 admits a unique multiplication, which is induced from A[z;α, δ], such that the spaceO(C, A) becomes an Arens-Michael algebra, which is denoted by O(C, A; α, δ),

2 and, moreover,A[z;α, δ] ' O(C, A; α, δ).




O(C, A) =

{f =

∞∑k=0

akzk :

∞∑k=0

‖ak‖λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

}






O(C, A) =

{f =

∞∑k=0

akzk :

∞∑k=0

‖ak‖λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

}






O(C, A) =

{f =

∞∑k=0

akzk :

∞∑k=0

‖ak‖λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

}




The similar theorem can be formulated for Laurent Ore extensions.

Theorem (A. Pirkovskii)Let A be an algebra, α ∈ Aut(A), such that (α, α−1) is an m-localizable family.Fix a generating family of seminorms {‖·‖λ : λ ∈ Λ} on A. Then the locallyconvex space

O(C∗, A) ={ ∞∑k=−∞

akzk :

∞∑k=−∞

‖ak‖λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

}

1 admits a unique multiplication, which is induced from A[z, z−1;α], such thatthe space O(C∗, A) becomes an Arens-Michael algebra, which as denoted byO(C∗, A; α)

2 and, moreover,A[z, z−1;α] ' O(C∗, A; α).




O(C∗, A) ={ ∞∑k=−∞

akzk :

∞∑k=−∞

‖ak‖λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

}






O(C∗, A) ={ ∞∑k=−∞

akzk :

∞∑k=−∞

‖ak‖λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

}






O(C∗, A) ={ ∞∑k=−∞

akzk :

∞∑k=−∞

‖ak‖λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

}




DefinitionSuppose that A is an Arens-Michael algebra and M is an A-⊗-bimodule (themodule action is jointly continuous). Fix a generating family of seminorms{‖·‖λ : λ ∈ Λ} on M . Then we can define the following locally convex space:

TA(M) = A⊕

{(xk) ∈

∞∏k=1

M ⊗k :∞∑k=1‖xk‖⊗kλ ρk <∞, λ ∈ Λ, 0 < ρ <∞

},

where‖·‖⊗kλ = ‖·‖λ ⊗π · · · ⊗π ‖·‖λ︸︷︷︸

k times

Theorem (A. Pirkovskii)1 TA(M) admits a unique multiplication, which is induced from TA(M), such

that TA(M) becomes an Arens-Michael algebra.2 For any algebra A and α ∈ End(A) we have

A[t;α] ' TA(Aα) ' TA(Aα).



TA(M) = A⊕

{(xk) ∈

∞∏k=1

M ⊗k :∞∑k=1‖xk‖⊗kλ ρk <∞, λ ∈ Λ, 0 < ρ <∞

},

where‖·‖⊗kλ = ‖·‖λ ⊗π · · · ⊗π ‖·‖λ︸︷︷︸

k times


that TA(M) becomes an Arens-Michael algebra.

2 For any algebra A and α ∈ End(A) we have




TA(M) = A⊕

{(xk) ∈

∞∏k=1

M ⊗k :∞∑k=1‖xk‖⊗kλ ρk <∞, λ ∈ Λ, 0 < ρ <∞

},

where‖·‖⊗kλ = ‖·‖λ ⊗π · · · ⊗π ‖·‖λ︸︷︷︸

k times


that TA(M) becomes an Arens-Michael algebra.2 For any algebra A and α ∈ End(A) we have



More ”explicit” description of TA(Aα)For any Arens-Michael algebra A and α ∈ End(A) the algebra TA(Aα) can berepresented as a non-commutative power series algebra. To do this, let us considerthe canonical isomorphism

in : (Aα)⊗n ' Aαn , in(a1 ⊗ · · · ⊗ an) = a1α(a2)α2(a3) . . . αn−1(an).

Fix a generating family of seminorms {|| · ||λ : λ ∈ Λ} on A.

Proposition

TA(Aα) '{ ∞∑k=0

akxk :

∞∑k=0‖ak‖(k)

λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

},

where‖a‖(k)

λ = infa=∑

kin(ak1⊗···⊗akn)

∑k

‖ak1‖λ . . . ‖akn‖λ ,

the inf is taken w.r.t the set of all representations

a =m∑k=1

in(ak1 ⊗ · · · ⊗ akn).





Proposition

TA(Aα) '{ ∞∑k=0

akxk :

∞∑k=0‖ak‖(k)

λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

},

where‖a‖(k)

λ = infa=∑


∑k

‖ak1‖λ . . . ‖akn‖λ ,


a =m∑k=1

in(ak1 ⊗ · · · ⊗ akn).





Proposition

TA(Aα) '{ ∞∑k=0

akxk :

∞∑k=0‖ak‖(k)

λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

},

where‖a‖(k)

λ = infa=∑


∑k

‖ak1‖λ . . . ‖akn‖λ ,


a =m∑k=1

in(ak1 ⊗ · · · ⊗ akn).





Proposition

TA(Aα) '{ ∞∑k=0

akxk :

∞∑k=0‖ak‖(k)

λ ρk <∞, λ ∈ Λ, 0 < ρ <∞

},

where‖a‖(k)

λ = infa=∑


∑k

‖ak1‖λ . . . ‖akn‖λ ,


a =m∑k=1

in(ak1 ⊗ · · · ⊗ akn).


DefinitionLet A be an algebra and M be an A-bimodule. Then M is an invertibleA-bimodule if there exist an A-bimodule M−1 and two A-bimodule topologicalisomorphisms i1 : M ⊗AM−1 ' A and i2 : M−1 ⊗AM ' A (convolutions)satisfying the associativity property:

M ⊗AM−1 ⊗AM M ⊗A A

A⊗AM M

IdM⊗i2

i1⊗IdM m⊗r→mr

r⊗m→rm

M−1 ⊗AM ⊗AM−1 M−1 ⊗A A

A⊗AM−1 M−1

IdM⊗i2

i1⊗IdM n⊗r→nr

r⊗n→rn

With every invertible A-bimodule M one can associate the following algebra:

LA(M) =⊕n∈Z

M⊗n, M⊗−k = (M−1)⊗k.




A⊗AM M

IdM⊗i2

i1⊗IdM m⊗r→mr

r⊗m→rm

M−1 ⊗AM ⊗AM−1 M−1 ⊗A A

A⊗AM−1 M−1

IdM⊗i2

i1⊗IdM n⊗r→nr

r⊗n→rn


LA(M) =⊕n∈Z

M⊗n, M⊗−k = (M−1)⊗k.




A⊗AM M

IdM⊗i2

i1⊗IdM m⊗r→mr

r⊗m→rm

M−1 ⊗AM ⊗AM−1 M−1 ⊗A A

A⊗AM−1 M−1

IdM⊗i2

i1⊗IdM n⊗r→nr

r⊗n→rn


LA(M) =⊕n∈Z

M⊗n, M⊗−k = (M−1)⊗k.


DefinitionLet A be an Arens-Michel algebra and M be an A-⊗-bimodule. Then we call Man analytically invertible A-⊗-bimodule if there exist an A-⊗-bimodule M−1 andtwo A-⊗-bimodule topological isomorphisms i1 : M⊗AM−1 ' A andi2 : M−1⊗AM ' A, satisfying the associativity property, as in the algebraicdefinition.

ExampleFor any Arens-Michael algebra A and α ∈ Aut(A) the bimodules Aα and Aα−1

are inverse and analytically inverse A-⊗-bimodules.


DefinitionLet A be an Arens-Michel algebra and M be an A-⊗-bimodule. Then we call Man analytically invertible A-⊗-bimodule if there exist an A-⊗-bimodule M−1 andtwo A-⊗-bimodule topological isomorphisms i1 : M⊗AM−1 ' A andi2 : M−1⊗AM ' A, satisfying the associativity property, as in the algebraicdefinition.

ExampleFor any Arens-Michael algebra A and α ∈ Aut(A) the bimodules Aα and Aα−1

are inverse and analytically inverse A-⊗-bimodules.


DefinitionSuppose that A is a Frechet-Arens-Michael algebra and M,M−1 is a pair ofanalytically invertible Frechet A-⊗-bimodules. Then we define

LA(M) := TA(M ⊕M−1)/I,

where I is the smallest closed two-sided ideal, containing(0, 0, x⊗ y, 0, . . . )− (i1(x⊗ y), 0, 0, . . . ) and(0, 0, y ⊗ x, 0, . . . )− (i2(y ⊗ x), 0, 0, . . . ) for any x ∈M , y ∈M−1.

Theorem (A. Pirkovskii, P. Kosenko)If A is the algebra with Arens-Michael envelope which is Frechet, andα ∈ Aut(A), then the following isomorphism takes place:

A[t, t−1;α] ' LA(Aα) ' LA(Aα)


DefinitionSuppose that A is a Frechet-Arens-Michael algebra and M,M−1 is a pair ofanalytically invertible Frechet A-⊗-bimodules. Then we define

LA(M) := TA(M ⊕M−1)/I,

where I is the smallest closed two-sided ideal, containing(0, 0, x⊗ y, 0, . . . )− (i1(x⊗ y), 0, 0, . . . ) and(0, 0, y ⊗ x, 0, . . . )− (i2(y ⊗ x), 0, 0, . . . ) for any x ∈M , y ∈M−1.

Theorem (A. Pirkovskii, P. Kosenko)If A is the algebra with Arens-Michael envelope which is Frechet, andα ∈ Aut(A), then the following isomorphism takes place:

A[t, t−1;α] ' LA(Aα) ' LA(Aα)


The quantum enveloping algebra of sl2DefinitionThe quantum universal enveloping algebra Uq(sl2) is an associative unital algebragenerated by E, F , K, K−1 with the following relations:

KE = q2EK, KF = q−2FK, [E,F ] = K −K−1

q − q−1 .

When |q| = 1, we consider the following isomorphism:Uq(sl2) ' C[K,K−1][F ;α0][E;α1, δ].

All morphisms turn out to be m-localizable, so we get the following result:

Theorem (D. Pedchenko)Consider |q| = 1, q 6= −1, 1. Then

Uq(sl2) '

f =∑

i,j∈Z≥0k∈Z

cijkKiF jEk : ‖f‖ρ :=

∑i,j,k

|cijk|ρi+j+k <∞ ∀ρ > 0

(1)



KE = q2EK, KF = q−2FK, [E,F ] = K −K−1

q − q−1 .




Uq(sl2) '

f =∑

i,j∈Z≥0k∈Z


∑i,j,k


(1)



KE = q2EK, KF = q−2FK, [E,F ] = K −K−1

q − q−1 .




Uq(sl2) '

f =∑

i,j∈Z≥0k∈Z


∑i,j,k


(1)


When |q| 6= 1, the representation of Uq(sl2) as an iterated Ore extension is of nouse to us because the morphisms cease to be m-localizable, so we need toconsider another approach. Notice that

Uq(sl2) ' C 〈E,F 〉 [K,K−1;α]([E,F ]− K−K−1

q−q−1

) .So it suffices to describe LR(Rα) in the case when

R = F2 = C 〈E,F 〉, α(E) = q2E,α(F ) = q−2F.

However, we still do not know whether this algebra can be represented as analgebra consisting of (non-commutative) Laurent power series.


Examples1 Suppose that A = C(R) or O(C), α(f)(x) = f(x− 1). Then

TA(Aα) ' A[[x]], LA(Aα) = 0.

2 Suppose that A = O(C), and βq(f)(z) = f(qz). Then

TA(Aβq ) =

{f =

∑k≥0

akxk : ‖f‖λ,ρ =

∑k≥0

(‖ak‖λ|q|−k )ρk <∞ ∀0 < λ, ρ <∞

}

when |q| ≥ 1 and

TA(Aβq ) =

{f =

∑k≥0

akxk : ‖f‖λ,ρ =

∑k≥0

(‖ak‖λ|q|k )ρk <∞ ∀0 < λ, ρ <∞

}

when |q| ≤ 1, where ‖f‖ρ = sup|z|≤ρ

|f(z)| or

‖f‖ρ =

∥∥∥∥∥∞∑k=0

akzk

∥∥∥∥∥ρ

=∞∑k=0

|ak|ρk.


References

Dmitry Pedchenko. “Arens-Michael envelopes of Jordanian plane andUq(sl2)”. MA thesis. National Research University Higher School ofEconomics, 2015.A. Yu. Pirkovskii. “Arens-Michael envelopes, homological epimorphisms,and relatively quasi-free algebras”. In: Transactions of the MoscowMathematical Society 69 (2008).

J. L. Taylor. “A general framework for a multi-operator functionalcalculus”. In: Advances in Mathematics 9.2 (1972), pp. 183–252.


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Arens-Michael envelopes of Ore extensions · Peter Kosenko (Higher School of Economics) Arens-Michael envelopes of Ore extensions July 11, 2017 5 / 18 The Arens-Michael envelope is