- Email: [email protected]
Algebraically determined semidirect products
Topology and its Applications 175 (2014) 43–48
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Algebraically determined semidirect products We’am M. Al-Tameemi a,1 , Robert R. Kallman b,2 a
Texas A&M International University, Department of Engineering, Mathematics, and Physics, College of Arts and Sciences, LBVSC 312, 5201 University Boulevard, Laredo, TX 78041, USA b University of North Texas, Department of Mathematics, 1155 Union Circle 311430, Denton, TX 76203-5017, USA
a r t i c l e
i n f o
Article history: Received 20 August 2013 Accepted 28 June 2014 Available online 21 July 2014 MSC: primary 22F99 secondary 03E15, 46L05 Keywords: Polish topological groups Semidirect products Descriptive set theory Hilbert space Unitary operators C ∗ -algebras Analytic set
a b s t r a c t Let G be a Polish (i.e., complete separable metric topological) group. Define G to be an algebraically determined Polish group if given any Polish group L and an algebraic isomorphism ϕ : L → G, then ϕ is a topological isomorphism. The purpose of this paper is to prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This general theorem will provide a flowchart or recipe for proving that some special semidirect products are algebraically determined. For example, it may be used to prove that the natural semidirect product H G, where H is the additive group of a separable Hilbert space and G is a Polish group of unitaries on H acting transitively on the unit sphere with −I ∈ G, is algebraically determined. An example of such a G is the unitary group of a separable irreducible C ∗ -algebra with identity on H. Not all nontrivial semidirect products of Polish groups are algebraically determined, for it is known that the Heisenberg group H3 (R) is a semidirect product of the form R2 θ R1 and is not an algebraically determined Polish group. Published by Elsevier B.V.
1. Introduction Let G be a Polish (i.e., complete separable metric topological) group. Define G to be an algebraically determined Polish group if given any Polish group L and an algebraic isomorphism ϕ : L → G, then ϕ is a topological isomorphism. The study of algebraically determined Polish groups dates back to the work of E. Cartan, B.L. van der Waerden and H. Freudenthal, among many others. See the introduction to [1] for a condensed history of algebraically determined Polish groups and why they are of scientific interest. The purpose of this paper is to prove a general result (Theorem 4) that gives a useful list of steps which suffice to prove that a semidirect product of two Polish groups is algebraically determined. This general theorem will provide a flowchart or recipe for proving that some special semidirect products are algebraically
1 2
E-mail addresses: [email protected] (W.M. Al-Tameemi), [email protected] (R.R. Kallman). Tel.: +1 956 326 2440; fax: +1 956 326 2439. Tel.: +1 940 565 3329; fax: +1 940 565 4805.
http://dx.doi.org/10.1016/j.topol.2014.06.016 0166-8641/Published by Elsevier B.V.
44
W.M. Al-Tameemi, R.R. Kallman / Topology and its Applications 175 (2014) 43–48
determined and is an abstraction and simplification of the techniques introduced in [1] and [4]. Theorem 4 is of a general character and the steps needed to verify its hypotheses in any particular instance can be quite difficult and require considerable ingenuity. That such individual ingenuity is required can be seen from a few examples. Observe that R2 = R × R is a trivial semidirect product that is not algebraically determined. Furthermore, not all nontrivial semidirect products of Polish groups are algebraically determined, for it is known that the Heisenberg group H3 (R), consisting of all 3 × 3 upper triangular matrices with 1’s along the diagonal, is a semidirect product of the form R2 θ R1 and is not an algebraically determined Polish group [4]. Kechris and Rosendal [6] introduced the notion of Polish groups with ample generics. This is an important but very special class of Polish groups since any homomorphism of such a Polish group into a separable topological group is automatically continuous. Obviously any Polish group with ample generics is algebraically determined. The converse is false since it is easy to see that any connected Lie group cannot have ample generics even though there are many examples of Lie groups that are algebraically determined. For example, the known result that the real ax + b group, a very elementary semidirect product, is algebraically determined follows easily from Theorem 4. However, the real ax + b group cannot have ample generics since its natural injection into the complex ax + b group followed by a discontinuous automorphism of C is a discontinuous homomorphism into a separable group. Therefore Theorem 4 applies to a much wider class of Polish groups than merely those with ample generics. Some new results are easy consequences of Theorem 4. For example, it may be used to prove that the natural semidirect product HG, where H is the additive group of a separable Hilbert space and G is a Polish group of unitaries on H acting transitively on the unit sphere with −I ∈ G, is algebraically determined. An example of such a G is the unitary group U(A) of a separable irreducible C ∗ -algebra A with identity acting on H is algebraically determined. The mathematical tools used in this paper are descriptive set theory methods. Basic references can be found in [2,5,7] and [8]. 2. The mathematical tools If X is a topological space, let B(X), the Borel subsets of X, be the σ-algebra generated by the open subsets of X and let BP(X), the subsets of X with the Baire property, be the σ-algebra generated by B(X) and the meager subsets of X. If X is a Polish space, then BP(X) contains the analytic subsets of X. The following proposition, a slight paraphrase of Theorem 1.2.6 in [2], is the basic general principal used to establish the results discussed in this paper. It is a consequence of the Banach–Kuratowski–Pettis theorem (Theorem 9.09, [5]). Proposition 1. Let H and G be two Polish groups and let ϕ : H → G be an algebraic homomorphism that is BP(H)-measurable. Then ϕ is continuous. Furthermore, ϕ is open if ϕ(H) is nonmeager. In particular, if ϕ is an algebraic isomorphism, then ϕ is a topological isomorphism. The following proposition will prove to be most useful. Proposition 2. (Proposition 5, [1]) Let G be a Polish topological group, A ⊂ G an analytic subset and H ⊂ G an analytic subgroup such that A intersects each H-coset in exactly one point and G = AH. Then H is closed in G. 3. The main result Recall that if K and Q are groups and θ : Q → Aut(K), q → θq , then K θ Q, the semidirect product of K and Q, is the group whose underlying set is K × Q and whose multiplication is (k1 , q1 )(k2 , q2 ) =
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(k1 θq1 (k2 ), q1 q2 ). Semidirect products arise naturally in many mathematical and physical situations. If the θ is known and fixed in a given context, then K θ Q is very often abbreviated to K Q. Semidirect products are described in detail in [9]. Lemma 3. Let K and Q be two Polish groups and let θ : Q → Aut(K), q → θq , be a group homomorphism that satisfies K × Q → K, (k, q) → θq (k) is continuous. Then K θ Q is a Polish group in the product topology. Proof. K θ Q is a Polish space if it is given the product topology. If (k1 , q1 ), (k2 , q2 ) ∈ K θ Q, then (k1 , q1 )(k2 , q2 )−1 = (k1 θq1 q2−1 (k2−1 ), q1 q2−1 ) is continuous in the (k1 , q1 ) and (k2 , q2 ) variables. Hence, K θ Q is a topological group and therefore a Polish group. 2 The following theorem is the main result of this paper. It provides a very general road map or recipe on how to go about proving that various semidirect products are algebraically determined and helps avoid needless repetitive arguments. Some of the hypotheses appear fairly strong but they are weak enough to be applicable to just about every case of interest, as illustrated by the applications given in Section 4. Verifying the hypotheses in any individual case can require a considerable amount of ingenuity dependent on the algebraic structure of the semidirect products. For example, it will be shown in a future publication that the natural semidirect products Rn × GL(n, R) and Rn × SL(n, R) (n ≥ 2) are algebraically determined by verifying the hypotheses of Theorem 4, though this verification appears to require quite elaborate arguments, especially if n ≥ 2 is odd in the SL(n, R) case. Such a result of course is false if R is replaced with C. Proving such theorems is a rather delicate process since the natural semidirect product R3 SO(3, R) is not algebraically determined. Theorem 4. Let K and Q be two Polish groups and let θ : Q → Aut(K) be a group homomorphism that satisfies K × Q → K, (k, q) → θq (k) is continuous. Then K θ Q is a Polish group in the product topology. Let L be a Polish group and let ϕ : L → K θ Q be a group isomorphism. If ϕ−1 (K) and ϕ−1 (Q) are both analytic subgroups of L, then both ϕ−1 (K) and ϕ−1 (Q) are closed subgroups of L. Next, if, in addition, ϕ|ϕ−1 (K) : ϕ−1 (K) → K is measurable with respect to BP(ϕ−1 (K)), then ϕ|ϕ−1 (K) is a topological isomorphism. Furthermore, if, in addition, θ is injective, then ϕ|ϕ−1 (Q) : ϕ−1 (Q) → Q is a topological isomorphism. Finally, under all of these conditions, ϕ : L → K θ Q is a topological isomorphism and thus K θ Q is an algebraically determined Polish group. Proof. K θ Q is a Polish group by Lemma 3. If ϕ−1 (K) and ϕ−1 (Q) are both analytic subgroups of L, then they are both closed subgroups of L by Proposition 2. If ϕ|ϕ−1 (K) : ϕ−1 (K) → K is measurable with respect to BP(ϕ−1 (K)), then ϕ|ϕ−1 (K) is a topological isomorphism by Proposition 1. Next, suppose that θ is injective. Let {k }≥1 be dense in K. Then ψ : ϕ−1 ((e, q)) → ≥1 ϕ−1 ((e, q)) · ϕ−1 ((k , e)) · ϕ−1 ((e, q))−1 = ≥1 ϕ−1 ((θq (k ), e)), ψ : ϕ−1 (Q) → ≥1 ϕ−1 (K) is continuous and oneto-one since θ is injective. The range of ψ is a Borel set and ψ is a Borel isomorphism onto its range by Souslin’s theorem [7]. υ : ≥1 ϕ−1 ((θq (k ), e)) → ≥1 (θq (k ), e), υ : ψ(ϕ−1 (Q)) → ≥1 K is continuous, one-to-one and therefore has Borel range and is a Borel isomorphism onto its range by Souslin’s theorem [7]. Therefore υ ◦ ψ is a Borel isomorphism onto its range. On the other hand the mapping β : Q → ≥1 K, (e, q) → ≥1 (θq (k ), e) is continuous and one-to-one and therefore is a Borel isomorphism onto its range (υ ◦ ψ)(ϕ−1 (Q)) by Souslin’s theorem [7]. Therefore β −1 : ≥1 (θq (k ), e) → (e, q) is a Borel mapping. Hence β −1 ◦ υ ◦ ψ : ϕ−1 ((e, q)) → (e, q), ϕ−1 (Q) → Q is a Borel mapping. Therefore ϕ|ϕ−1 (Q) is a topological isomorphism by Proposition 1. The mapping f : ϕ−1 (K) × ϕ−1 (Q) → L, f : ϕ−1 (k, e) , ϕ−1 (e, q) → ϕ−1 (k, e) · ϕ−1 (e, q) = ϕ−1 (k, q)
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is continuous, one-to-one and onto. Hence f is a Borel isomorphism and its inverse, f −1 : ϕ−1 ((k, q)) → (ϕ−1 ((k, e)), ϕ−1 ((e, q))), is Borel isomorphism by Souslin’s theorem [7]. Then ϕ−1 (k, q) → ϕ−1 (k, e) , ϕ−1 (e, q) → (k, e), (e, q) → (k, e) · (e, q) = (k, q) is a Borel mapping. Then ϕ is a Borel mapping between Polish groups and hence ϕ is topological isomorphism by Proposition 1. Therefore K θ Q is an algebraically determined Polish group. 2 A few simple examples may indicate the technical delicacy required in examining even elementary semidirect products. The real ax +b group is an example of an algebraically determined Polish group where neither the K nor the Q is algebraically determined. On the other hand the natural semidirect product R3 SO(3, R) is not algebraically determined since it has a discontinuous automorphism (Tits [10]) even though SO(3, R), like any compact simple Lie group, is algebraically determined. Thus the fact that Q is algebraically determined has no bearing on whether or not K ϕ Q is algebraically determined. 4. Applications The main goal in this section is to prove the following theorem. Theorem 5. Let H be a separable complex Hilbert space, viewed as an additive Polish group. Let G be an abstract subgroup of U(H), the unitary group of H, that acts transitively on the unit sphere of H and contains −I. Suppose in addition that G is a Polish group in some topology such that H × G → H, (x, U ) → U (x), is continuous. Then the natural semidirect product H G is an algebraically determined Polish group. The proof of this theorem, which is done in a sequence of simple lemmas, is modeled on the proof of Theorem 27 of [1] for the special case G = U(H). The proof of the following lemma is an elementary computation. Lemma 6. The centralizer of (0, −I) in H G is G and H is a maximal abelian subgroup of H G. Lemma 7. Let K be real or complex inner product space with dim(K) ≥ 2 if K is real or dim(K) ≥ 1 if K is complex. Let x ∈ K with x ≤ 2. Then there exist y, z ∈ K with y = z = 1 and y + z = x. Proof. It suffices to consider the case in which K is an inner product space over the reals, for if K is complex, just restrict the scalars to the reals and replace the complex inner product with its real part. Then K will be a real inner product space and the real dimension of K will be at least two. Let x ∈ K satisfying x ≤ 2. Choose a unit vector v orthogonal to x and take y = x+av and z = x−av 2 2 , x+av 2 1 2 12 2 2 2 where a = (4 −x ) . Then y+z = x and a = (4 −x ). Therefore y = 2 = 4 (x2 +a2 v2 ) = 1 2 2 4 (x + (4 − x )) = 1. Thus y = 1. Similarly one can prove that z = 1. 2 Lemma 8. Let L be a Polish group and let ϕ : L → H G be an algebraic isomorphism. Then ϕ−1 (G) and ϕ−1 (H) are closed in L. Proof. Use Lemma 6. ϕ−1 (G) is the centralizer of ϕ−1 ((0, −I)) in L and therefore is closed in L. ϕ−1 (H) is maximal abelian in L and therefore is closed in L. 2 Lemma 9. Let L be a Polish group and let ϕ : L → H G be an algebraic isomorphism. Let δ > 0 and let A = {(z, I) | z ≤ δ, z ∈ H}. Then ϕ−1 (A) is analytic in ϕ−1 (H).
W.M. Al-Tameemi, R.R. Kallman / Topology and its Applications 175 (2014) 43–48
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Proof. Fix x0 ∈ H with x0 = 2δ . The mapping −1 ϕ−1 (0, U ) → ϕ−1 (0, U ) ϕ−1 (x0 , I) ϕ−1 (0, U ) = ϕ−1 U (x0 ), I , ϕ−1 (G) → ϕ−1 (H), is continuous. Therefore the range of this mapping, namely ϕ−1 ({(x, I) | x = δ/2}), is an analytic set in ϕ−1 (H) since ϕ−1 (G) is closed in L. ϕ−1 ({(x, I) | x = δ/2}) × ϕ−1 ({(y, I) | y = δ/2}) ⊆ ϕ−1 (H) × ϕ−1 (H) is also analytic since the product of two analytic sets is analytic. Therefore the mapping
ϕ−1 (x, I) , ϕ−1 (y, I) → ϕ−1 (x, I) · ϕ−1 (y, I) = ϕ−1 (x + y, I) , ϕ−1 (x, I) | x = δ/2 × ϕ−1 (y, I) | y = δ/2 → ϕ−1 (H)
is continuous and therefore has analytic range. But the range of this mapping is ϕ−1({(z, I) | z ≤ δ}), as follows from Lemma 7 by scaling by 2/δ. 2 Lemma 10. ϕ|ϕ−1 (H) is measurable with respect to BP(ϕ−1 (H)). Proof. Let BH ((0, I), δ) = {(x, I) | x < δ}. Then ϕ−1 (BH ((0, I), δ)) = n≥1 ϕ−1 ({(x, I) | x ≤ δ− n1 }) is analytic in ϕ−1 (H). Fix x0 ∈ H. Then ϕ−1 ({(x, I) | x − x0 < δ}) = ϕ−1 ((x0 , I)) + ϕ−1 ({(x, I) | x < δ}) is analytic since w → ϕ−1 ((x0 ), I) + w, H → H, is continuous. Let O be open in H such that O = −1 (O) = n≥1 ϕ−1 ({(x, I) | x − xn < δn }) is analytic in ϕ−1 (H). Hence n≥1 BH ((xn , I), δn ), then ϕ ϕ|ϕ−1 (H) is measurable with respect to BP(ϕ−1 (H)) since analytic subsets of ϕ−1 (H) are in BP(ϕ−1 (H)) (Corollary 1, p. 482, [7]). 2 Proof of Theorem 5. Apply the previous sequence of lemmas to Theorem 4. 2 Corollary 11. Let A be a separable irreducible concrete C ∗ -algebra with identity on the complex Hilbert space H and let U(A) be the group of unitaries in A. Then the natural semidirect product H U(A) is an algebraically determined Polish group. Proof. It is a simple exercise to check that H is necessarily separable under the hypotheses on A. Theorem 5 will imply this corollary if U(A) acts transitively on the unit sphere of H. This follows from a result of Glimm and Kadison [3]. 2 Corollary 12. (Theorem 27, [1]) Let U(H) be the group of unitaries on an infinite separable complex Hilbert space H with the strong operator topology. Then H U(H) is an algebraically determined Polish group. Proof. It is simple to check that HU(H) is a Polish group, that −I ∈ U(H) and that U(H) acts transitively on the unit sphere of H. Now apply Theorem 5. 2 References [1] A.G. Atim, Robert R. Kallman, The infinite unitary and related groups are algebraically determined Polish groups, Topol. Appl. 159 (12) (August 2012) 2831–2840, http://dx.doi.org/10.1016/j.topol.2012.04.018. [2] Howard Becker, Alexander S. Kechris, The Descriptive Set Theory of Polish Group Actions, Cambridge University Press, Cambridge, United Kingdom, ISBN 0-521-57605-9, 1996. [3] James G. Glimm, Richard V. Kadison, Unitary operators in C ∗ -algebras, Pac. J. Math. 10 (1960) 547–556. [4] Robert R. Kallman, Alexander P. McLinden, The Poincaré and related groups are algebraically determined Polish groups, Collect. Math. 61 (3) (2010) 337–352.
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[5] Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. [6] Alexander S. Kechris, Christian Rosendal, Turbulence, amalgamation and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc. 94 (2) (1997) 302–350. [7] Kazimierz Kuratowski, Topology, vol. 1, Academic Press, New York, 1966. [8] George W. Mackey, Borel structure in groups and their duals, Trans. Am. Math. Soc. 85 (1) (May 1957) 134–165. [9] Joseph J. Rotman, An Introduction to the Theory of Groups, third edition, William C. Brown Publishers, Dubuque, Iowa, ISBN 0-697-06882-X, 1988. [10] Jacques Tits, Homomorphismes “abstraits” de groupes de Lie, in: Symposia Mathematica, vol. XIII, Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972, Academic Press, New York, 1974, pp. 479–499.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Algebraically determined semidirect products We’am M. Al-Tameemi a,1 , Robert R. Kallman b,2 a
Texas A&M International University, Department of Engineering, Mathematics, and Physics, College of Arts and Sciences, LBVSC 312, 5201 University Boulevard, Laredo, TX 78041, USA b University of North Texas, Department of Mathematics, 1155 Union Circle 311430, Denton, TX 76203-5017, USA
a r t i c l e
i n f o
Article history: Received 20 August 2013 Accepted 28 June 2014 Available online 21 July 2014 MSC: primary 22F99 secondary 03E15, 46L05 Keywords: Polish topological groups Semidirect products Descriptive set theory Hilbert space Unitary operators C ∗ -algebras Analytic set
a b s t r a c t Let G be a Polish (i.e., complete separable metric topological) group. Define G to be an algebraically determined Polish group if given any Polish group L and an algebraic isomorphism ϕ : L → G, then ϕ is a topological isomorphism. The purpose of this paper is to prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This general theorem will provide a flowchart or recipe for proving that some special semidirect products are algebraically determined. For example, it may be used to prove that the natural semidirect product H G, where H is the additive group of a separable Hilbert space and G is a Polish group of unitaries on H acting transitively on the unit sphere with −I ∈ G, is algebraically determined. An example of such a G is the unitary group of a separable irreducible C ∗ -algebra with identity on H. Not all nontrivial semidirect products of Polish groups are algebraically determined, for it is known that the Heisenberg group H3 (R) is a semidirect product of the form R2 θ R1 and is not an algebraically determined Polish group. Published by Elsevier B.V.
1. Introduction Let G be a Polish (i.e., complete separable metric topological) group. Define G to be an algebraically determined Polish group if given any Polish group L and an algebraic isomorphism ϕ : L → G, then ϕ is a topological isomorphism. The study of algebraically determined Polish groups dates back to the work of E. Cartan, B.L. van der Waerden and H. Freudenthal, among many others. See the introduction to [1] for a condensed history of algebraically determined Polish groups and why they are of scientific interest. The purpose of this paper is to prove a general result (Theorem 4) that gives a useful list of steps which suffice to prove that a semidirect product of two Polish groups is algebraically determined. This general theorem will provide a flowchart or recipe for proving that some special semidirect products are algebraically
1 2
E-mail addresses: [email protected] (W.M. Al-Tameemi), [email protected] (R.R. Kallman). Tel.: +1 956 326 2440; fax: +1 956 326 2439. Tel.: +1 940 565 3329; fax: +1 940 565 4805.
http://dx.doi.org/10.1016/j.topol.2014.06.016 0166-8641/Published by Elsevier B.V.
44
W.M. Al-Tameemi, R.R. Kallman / Topology and its Applications 175 (2014) 43–48
determined and is an abstraction and simplification of the techniques introduced in [1] and [4]. Theorem 4 is of a general character and the steps needed to verify its hypotheses in any particular instance can be quite difficult and require considerable ingenuity. That such individual ingenuity is required can be seen from a few examples. Observe that R2 = R × R is a trivial semidirect product that is not algebraically determined. Furthermore, not all nontrivial semidirect products of Polish groups are algebraically determined, for it is known that the Heisenberg group H3 (R), consisting of all 3 × 3 upper triangular matrices with 1’s along the diagonal, is a semidirect product of the form R2 θ R1 and is not an algebraically determined Polish group [4]. Kechris and Rosendal [6] introduced the notion of Polish groups with ample generics. This is an important but very special class of Polish groups since any homomorphism of such a Polish group into a separable topological group is automatically continuous. Obviously any Polish group with ample generics is algebraically determined. The converse is false since it is easy to see that any connected Lie group cannot have ample generics even though there are many examples of Lie groups that are algebraically determined. For example, the known result that the real ax + b group, a very elementary semidirect product, is algebraically determined follows easily from Theorem 4. However, the real ax + b group cannot have ample generics since its natural injection into the complex ax + b group followed by a discontinuous automorphism of C is a discontinuous homomorphism into a separable group. Therefore Theorem 4 applies to a much wider class of Polish groups than merely those with ample generics. Some new results are easy consequences of Theorem 4. For example, it may be used to prove that the natural semidirect product HG, where H is the additive group of a separable Hilbert space and G is a Polish group of unitaries on H acting transitively on the unit sphere with −I ∈ G, is algebraically determined. An example of such a G is the unitary group U(A) of a separable irreducible C ∗ -algebra A with identity acting on H is algebraically determined. The mathematical tools used in this paper are descriptive set theory methods. Basic references can be found in [2,5,7] and [8]. 2. The mathematical tools If X is a topological space, let B(X), the Borel subsets of X, be the σ-algebra generated by the open subsets of X and let BP(X), the subsets of X with the Baire property, be the σ-algebra generated by B(X) and the meager subsets of X. If X is a Polish space, then BP(X) contains the analytic subsets of X. The following proposition, a slight paraphrase of Theorem 1.2.6 in [2], is the basic general principal used to establish the results discussed in this paper. It is a consequence of the Banach–Kuratowski–Pettis theorem (Theorem 9.09, [5]). Proposition 1. Let H and G be two Polish groups and let ϕ : H → G be an algebraic homomorphism that is BP(H)-measurable. Then ϕ is continuous. Furthermore, ϕ is open if ϕ(H) is nonmeager. In particular, if ϕ is an algebraic isomorphism, then ϕ is a topological isomorphism. The following proposition will prove to be most useful. Proposition 2. (Proposition 5, [1]) Let G be a Polish topological group, A ⊂ G an analytic subset and H ⊂ G an analytic subgroup such that A intersects each H-coset in exactly one point and G = AH. Then H is closed in G. 3. The main result Recall that if K and Q are groups and θ : Q → Aut(K), q → θq , then K θ Q, the semidirect product of K and Q, is the group whose underlying set is K × Q and whose multiplication is (k1 , q1 )(k2 , q2 ) =
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(k1 θq1 (k2 ), q1 q2 ). Semidirect products arise naturally in many mathematical and physical situations. If the θ is known and fixed in a given context, then K θ Q is very often abbreviated to K Q. Semidirect products are described in detail in [9]. Lemma 3. Let K and Q be two Polish groups and let θ : Q → Aut(K), q → θq , be a group homomorphism that satisfies K × Q → K, (k, q) → θq (k) is continuous. Then K θ Q is a Polish group in the product topology. Proof. K θ Q is a Polish space if it is given the product topology. If (k1 , q1 ), (k2 , q2 ) ∈ K θ Q, then (k1 , q1 )(k2 , q2 )−1 = (k1 θq1 q2−1 (k2−1 ), q1 q2−1 ) is continuous in the (k1 , q1 ) and (k2 , q2 ) variables. Hence, K θ Q is a topological group and therefore a Polish group. 2 The following theorem is the main result of this paper. It provides a very general road map or recipe on how to go about proving that various semidirect products are algebraically determined and helps avoid needless repetitive arguments. Some of the hypotheses appear fairly strong but they are weak enough to be applicable to just about every case of interest, as illustrated by the applications given in Section 4. Verifying the hypotheses in any individual case can require a considerable amount of ingenuity dependent on the algebraic structure of the semidirect products. For example, it will be shown in a future publication that the natural semidirect products Rn × GL(n, R) and Rn × SL(n, R) (n ≥ 2) are algebraically determined by verifying the hypotheses of Theorem 4, though this verification appears to require quite elaborate arguments, especially if n ≥ 2 is odd in the SL(n, R) case. Such a result of course is false if R is replaced with C. Proving such theorems is a rather delicate process since the natural semidirect product R3 SO(3, R) is not algebraically determined. Theorem 4. Let K and Q be two Polish groups and let θ : Q → Aut(K) be a group homomorphism that satisfies K × Q → K, (k, q) → θq (k) is continuous. Then K θ Q is a Polish group in the product topology. Let L be a Polish group and let ϕ : L → K θ Q be a group isomorphism. If ϕ−1 (K) and ϕ−1 (Q) are both analytic subgroups of L, then both ϕ−1 (K) and ϕ−1 (Q) are closed subgroups of L. Next, if, in addition, ϕ|ϕ−1 (K) : ϕ−1 (K) → K is measurable with respect to BP(ϕ−1 (K)), then ϕ|ϕ−1 (K) is a topological isomorphism. Furthermore, if, in addition, θ is injective, then ϕ|ϕ−1 (Q) : ϕ−1 (Q) → Q is a topological isomorphism. Finally, under all of these conditions, ϕ : L → K θ Q is a topological isomorphism and thus K θ Q is an algebraically determined Polish group. Proof. K θ Q is a Polish group by Lemma 3. If ϕ−1 (K) and ϕ−1 (Q) are both analytic subgroups of L, then they are both closed subgroups of L by Proposition 2. If ϕ|ϕ−1 (K) : ϕ−1 (K) → K is measurable with respect to BP(ϕ−1 (K)), then ϕ|ϕ−1 (K) is a topological isomorphism by Proposition 1. Next, suppose that θ is injective. Let {k }≥1 be dense in K. Then ψ : ϕ−1 ((e, q)) → ≥1 ϕ−1 ((e, q)) · ϕ−1 ((k , e)) · ϕ−1 ((e, q))−1 = ≥1 ϕ−1 ((θq (k ), e)), ψ : ϕ−1 (Q) → ≥1 ϕ−1 (K) is continuous and oneto-one since θ is injective. The range of ψ is a Borel set and ψ is a Borel isomorphism onto its range by Souslin’s theorem [7]. υ : ≥1 ϕ−1 ((θq (k ), e)) → ≥1 (θq (k ), e), υ : ψ(ϕ−1 (Q)) → ≥1 K is continuous, one-to-one and therefore has Borel range and is a Borel isomorphism onto its range by Souslin’s theorem [7]. Therefore υ ◦ ψ is a Borel isomorphism onto its range. On the other hand the mapping β : Q → ≥1 K, (e, q) → ≥1 (θq (k ), e) is continuous and one-to-one and therefore is a Borel isomorphism onto its range (υ ◦ ψ)(ϕ−1 (Q)) by Souslin’s theorem [7]. Therefore β −1 : ≥1 (θq (k ), e) → (e, q) is a Borel mapping. Hence β −1 ◦ υ ◦ ψ : ϕ−1 ((e, q)) → (e, q), ϕ−1 (Q) → Q is a Borel mapping. Therefore ϕ|ϕ−1 (Q) is a topological isomorphism by Proposition 1. The mapping f : ϕ−1 (K) × ϕ−1 (Q) → L, f : ϕ−1 (k, e) , ϕ−1 (e, q) → ϕ−1 (k, e) · ϕ−1 (e, q) = ϕ−1 (k, q)
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is continuous, one-to-one and onto. Hence f is a Borel isomorphism and its inverse, f −1 : ϕ−1 ((k, q)) → (ϕ−1 ((k, e)), ϕ−1 ((e, q))), is Borel isomorphism by Souslin’s theorem [7]. Then ϕ−1 (k, q) → ϕ−1 (k, e) , ϕ−1 (e, q) → (k, e), (e, q) → (k, e) · (e, q) = (k, q) is a Borel mapping. Then ϕ is a Borel mapping between Polish groups and hence ϕ is topological isomorphism by Proposition 1. Therefore K θ Q is an algebraically determined Polish group. 2 A few simple examples may indicate the technical delicacy required in examining even elementary semidirect products. The real ax +b group is an example of an algebraically determined Polish group where neither the K nor the Q is algebraically determined. On the other hand the natural semidirect product R3 SO(3, R) is not algebraically determined since it has a discontinuous automorphism (Tits [10]) even though SO(3, R), like any compact simple Lie group, is algebraically determined. Thus the fact that Q is algebraically determined has no bearing on whether or not K ϕ Q is algebraically determined. 4. Applications The main goal in this section is to prove the following theorem. Theorem 5. Let H be a separable complex Hilbert space, viewed as an additive Polish group. Let G be an abstract subgroup of U(H), the unitary group of H, that acts transitively on the unit sphere of H and contains −I. Suppose in addition that G is a Polish group in some topology such that H × G → H, (x, U ) → U (x), is continuous. Then the natural semidirect product H G is an algebraically determined Polish group. The proof of this theorem, which is done in a sequence of simple lemmas, is modeled on the proof of Theorem 27 of [1] for the special case G = U(H). The proof of the following lemma is an elementary computation. Lemma 6. The centralizer of (0, −I) in H G is G and H is a maximal abelian subgroup of H G. Lemma 7. Let K be real or complex inner product space with dim(K) ≥ 2 if K is real or dim(K) ≥ 1 if K is complex. Let x ∈ K with x ≤ 2. Then there exist y, z ∈ K with y = z = 1 and y + z = x. Proof. It suffices to consider the case in which K is an inner product space over the reals, for if K is complex, just restrict the scalars to the reals and replace the complex inner product with its real part. Then K will be a real inner product space and the real dimension of K will be at least two. Let x ∈ K satisfying x ≤ 2. Choose a unit vector v orthogonal to x and take y = x+av and z = x−av 2 2 , x+av 2 1 2 12 2 2 2 where a = (4 −x ) . Then y+z = x and a = (4 −x ). Therefore y = 2 = 4 (x2 +a2 v2 ) = 1 2 2 4 (x + (4 − x )) = 1. Thus y = 1. Similarly one can prove that z = 1. 2 Lemma 8. Let L be a Polish group and let ϕ : L → H G be an algebraic isomorphism. Then ϕ−1 (G) and ϕ−1 (H) are closed in L. Proof. Use Lemma 6. ϕ−1 (G) is the centralizer of ϕ−1 ((0, −I)) in L and therefore is closed in L. ϕ−1 (H) is maximal abelian in L and therefore is closed in L. 2 Lemma 9. Let L be a Polish group and let ϕ : L → H G be an algebraic isomorphism. Let δ > 0 and let A = {(z, I) | z ≤ δ, z ∈ H}. Then ϕ−1 (A) is analytic in ϕ−1 (H).
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Proof. Fix x0 ∈ H with x0 = 2δ . The mapping −1 ϕ−1 (0, U ) → ϕ−1 (0, U ) ϕ−1 (x0 , I) ϕ−1 (0, U ) = ϕ−1 U (x0 ), I , ϕ−1 (G) → ϕ−1 (H), is continuous. Therefore the range of this mapping, namely ϕ−1 ({(x, I) | x = δ/2}), is an analytic set in ϕ−1 (H) since ϕ−1 (G) is closed in L. ϕ−1 ({(x, I) | x = δ/2}) × ϕ−1 ({(y, I) | y = δ/2}) ⊆ ϕ−1 (H) × ϕ−1 (H) is also analytic since the product of two analytic sets is analytic. Therefore the mapping
ϕ−1 (x, I) , ϕ−1 (y, I) → ϕ−1 (x, I) · ϕ−1 (y, I) = ϕ−1 (x + y, I) , ϕ−1 (x, I) | x = δ/2 × ϕ−1 (y, I) | y = δ/2 → ϕ−1 (H)
is continuous and therefore has analytic range. But the range of this mapping is ϕ−1({(z, I) | z ≤ δ}), as follows from Lemma 7 by scaling by 2/δ. 2 Lemma 10. ϕ|ϕ−1 (H) is measurable with respect to BP(ϕ−1 (H)). Proof. Let BH ((0, I), δ) = {(x, I) | x < δ}. Then ϕ−1 (BH ((0, I), δ)) = n≥1 ϕ−1 ({(x, I) | x ≤ δ− n1 }) is analytic in ϕ−1 (H). Fix x0 ∈ H. Then ϕ−1 ({(x, I) | x − x0 < δ}) = ϕ−1 ((x0 , I)) + ϕ−1 ({(x, I) | x < δ}) is analytic since w → ϕ−1 ((x0 ), I) + w, H → H, is continuous. Let O be open in H such that O = −1 (O) = n≥1 ϕ−1 ({(x, I) | x − xn < δn }) is analytic in ϕ−1 (H). Hence n≥1 BH ((xn , I), δn ), then ϕ ϕ|ϕ−1 (H) is measurable with respect to BP(ϕ−1 (H)) since analytic subsets of ϕ−1 (H) are in BP(ϕ−1 (H)) (Corollary 1, p. 482, [7]). 2 Proof of Theorem 5. Apply the previous sequence of lemmas to Theorem 4. 2 Corollary 11. Let A be a separable irreducible concrete C ∗ -algebra with identity on the complex Hilbert space H and let U(A) be the group of unitaries in A. Then the natural semidirect product H U(A) is an algebraically determined Polish group. Proof. It is a simple exercise to check that H is necessarily separable under the hypotheses on A. Theorem 5 will imply this corollary if U(A) acts transitively on the unit sphere of H. This follows from a result of Glimm and Kadison [3]. 2 Corollary 12. (Theorem 27, [1]) Let U(H) be the group of unitaries on an infinite separable complex Hilbert space H with the strong operator topology. Then H U(H) is an algebraically determined Polish group. Proof. It is simple to check that HU(H) is a Polish group, that −I ∈ U(H) and that U(H) acts transitively on the unit sphere of H. Now apply Theorem 5. 2 References [1] A.G. Atim, Robert R. Kallman, The infinite unitary and related groups are algebraically determined Polish groups, Topol. Appl. 159 (12) (August 2012) 2831–2840, http://dx.doi.org/10.1016/j.topol.2012.04.018. [2] Howard Becker, Alexander S. Kechris, The Descriptive Set Theory of Polish Group Actions, Cambridge University Press, Cambridge, United Kingdom, ISBN 0-521-57605-9, 1996. [3] James G. Glimm, Richard V. Kadison, Unitary operators in C ∗ -algebras, Pac. J. Math. 10 (1960) 547–556. [4] Robert R. Kallman, Alexander P. McLinden, The Poincaré and related groups are algebraically determined Polish groups, Collect. Math. 61 (3) (2010) 337–352.
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[5] Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. [6] Alexander S. Kechris, Christian Rosendal, Turbulence, amalgamation and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc. 94 (2) (1997) 302–350. [7] Kazimierz Kuratowski, Topology, vol. 1, Academic Press, New York, 1966. [8] George W. Mackey, Borel structure in groups and their duals, Trans. Am. Math. Soc. 85 (1) (May 1957) 134–165. [9] Joseph J. Rotman, An Introduction to the Theory of Groups, third edition, William C. Brown Publishers, Dubuque, Iowa, ISBN 0-697-06882-X, 1988. [10] Jacques Tits, Homomorphismes “abstraits” de groupes de Lie, in: Symposia Mathematica, vol. XIII, Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972, Academic Press, New York, 1974, pp. 479–499.