LECTURE NOTES IN TOPOLOGICAL GROUPS (2014) UPDATED 7.2.17

MICHAEL MEGRELISHVILI

Contents 1. Lecture 1 1 2. Lecture 2 4 3. Lecture 3 8 4. Lecture 4 13 5. Lecture 5 17 6. Lectures 6 and 7 21 7. Lecture 8 27 8. Lecture 9 30 9. Lectures 10,11,12 34 References 43

1. Lecture 1 Definition 1.1. Let (G, m) be a and τ be a topology on G. We say that (G, m, τ) (or, simply, G) is a if the two basic operations m : G × G → G, (x, y) 7→ m(x, y) := xy and the inversion i : G → G, x 7→ x−1 are continuous. We say also that τ is a group topology on the group G. By TGr we denote the class of all topological groups. In most cases later we consider Hausdorff groups. Morphisms in TGr are continuous homomorphisms. For example, −1 id :(R, τdiscrete) → R is a continuous homomorphism but not id . Algebraically id is of course an isomorphism. That is, isomorphism in Gr but not in TGr. Note that every group with the discrete topology is a topological group. Hence, Gr ⊂ T Gr. Trivial topology is also a group topology on every group. We say that a topological group is Hausdorff, compact, metrizable, separable etc. if the given topology on G satisfies the corresponding topological property. Remarks 1.2.

Date: February, 2017. 1 2

(1) (a short version) 1.1 is equivalent to the following condition: G × G → G, (x, y) 7→ xy−1 is continuous. (2) (in terms of nbds) 1.1 is equivalent to the following conditions: (a) ∀U ∈ N(xy) ∃V ∈ N(x),W ∈ N(y): VW ⊂ U (b) ∀U ∈ N(x−1) ∃V ∈ N(x): V −1 ⊂ U. Note that (b) is equivalent to (b’) ∀U ∈ N(x−1) U −1 ∈ N(x). (3) (in terms of (generalized) sequences) If G is metrizable then 1.1 is equivalent to the following condition: −1 −1 xn → x, yn → y ⇒ xnyn → xy In general one may use the nets ( generalized sequences).

Exercise 1.3. (1) Show that R2 and R are nonhomeomorphic topological spaces but algebraically these groups are isomorphic. (2) Show that the discrete spaces Z and Z × Z2 are homeomorphic but as groups they are not isomorphic. Definition 1.4. (1) In terms of Definition 1.1 we say that G is: paratopological group if m is continuous. (2) Let (S, m) be a semigroup and τ be a topology on S. We say that the semi- group S is: (a) topological semigroup if m : S × S → S is continuous. So, a paratopolog- ical group is a topological semigroup. (b) semitopological if m is separately continuous. That is, if all left and right translations

la : S → S, x 7→ ax, ra : S → S, x 7→ xa are continuous for every a ∈ S. (c) right (left) topological if right (left) translations of S are continuous.

Example 1.5. (1) (para but not topo) Let τs be the Sorgenfrey topology (standard topological base is {[a, b)}) on the group (R, +) of all reals. Then (R, τs, +) is a paratopological but not topological group. Hint: [0, 1) ∈ τs but (−1, 0] ∈/ τs. (2) (semi but not para) For every group G the pair (G, τcof ) (with the cofinite topology) is a which satisfies T1. It is a paratopological group iff G is finite. (3) (right but not left) For every X consider the semigroup (XX , ◦) of all selfmaps wrt product (=pointwise) topology. Then XX is right X X topological. * If X ∈ T1 the teft translation lf : X → X is continuous iff f ∈ C(X,X). Note that if X is compact then XX is compact by the Tychonoff theorem. Theorem 1.6. Let S be a right topological semigroup. If S is compact then it contains at least one idempotent. 3

Proof. We have to show that there exists m ∈ S such that m2 = m. By Zorn’s Lemma (and compactness of S) thee exists a minimal compact sub- semigroup M ⊆ S (indeed, for any chain of compact subsemigroups the intersection is a nonempty compact subsemigroup (by the compactness) of S). Take arbitrary m ∈ M. Our aim is to show that m2 = m. Consider the set Mm. Then Mm ⊆ M is a subsemigroup. Observe also that Mm is compact. By the minimality of M we necessarily have Mm = M. This implies that um = m for some u ∈ S. Hence, the following set K := {x ∈ M : xm = m} is nonempty. Moreover, K is a subsemigroup and compact (again use the continuity of right translations). Since K ⊆ M we necessarily have K = M. This implies that m ∈ K. Therefore, m2 = m.  4

2. Lecture 2 Definition 2.1. A topological space X is said to be homogeneous if for every x, y ∈ X there exists an autohomeomorphism h : X → X (notation: h ∈ H(X)) s.t. h(x) = y. Lemma 2.2. For every semitopological group the left (right) translations are home- omorphism. For every topological group the inversion map is a homeomorphism.

−1 −1 −1 Proof. Observe that la = la−1 (ra = ra−1 ) and i = i.  Proposition 2.3. Every semitopological group (hence, also, every topological, as a topological space) is homogeneous. Definition 2.4. A topological space (X, τ) is said to be of group type if there exists a group structure m on X such that (X, τ, m) is a topological group. Remarks 2.5. (1) [0, 1]n for every natural n is not homogeneous hence not of group type. (2) The Hilbert cube [0, 1]N is homogeneous (Keller). At the same time it has the fixed point property: every continuous map h : [0, 1]N → [0, 1]N has a fixed point. It follows that the Hilbert cube is not of group type. Moreover, there is no structure of a left (right) topological group on it. (3) The Cantor set C ⊂ [0, 1] is of group type. Indeed, C is homeomorphic to the N topological space Z2 , which is a topological group. (4) The space of all irrational numbers R\Q is of group type being homeomorphic to ZN. (5) The Sorgenfrey line as a topological space is homogeneous but not of group type (up to a non-trivial theorem of Kakutani below: every Hausdorff topological group with the first countable property B1 is metrizable). Some examples of topological groups: (1) Every group in the discrete topology. (2) GLn(R) are locally compact metrizable topological group. It, as a metric 2 space, is embedded isometrically into the Euclidean Rn . (3) The On(R) ⊂ GLn(R) is compact by Heine-Borel thm being a bounded and a closed subset in the metric space GLn(R). Note that O2(R) is isomorphic in TGr with the ”” T ≤ C∗. (4) TGr is closed under: , factor-groups, topological products, box products. (5) Every Euclidean space Rn and Tn the n-dimensional torus. (6) Every normed space (more generally, any linear topological space). (7) (Z, dp) the wrt the p-adic metric. It is a precompact group (totally bounded in its metric) and its completion is the compact topological group of all p-adic integers. (8) For every K the group of all autohomeomorphisms H(K) en- dowed with the so-called compact-open topology (we define it later). (9) For every metric space (X, d) the group of all onto isometries Iso(X, d) ⊂ XX endowed with the pointwise topology inherited from XX . (10) For every Banach space (V, ||·||) the group Isolin(V ) of all linear onto isometries V → V endowed with the pointwise topology inherited from V V . For example, 5

n if V := R is the Euclidean space then Isolin(V ) = On(R) the orthogonal group. Note that, in contrast to the case of Rn, for infinite dimensional V the topological group Isolin(V ) as usual is not compact. Moreover, Teleman’s theorems show that every Hausdorff topological group G is embedded into Isolin(V ) for suitable V and also into some H(K) for suitable compact space K. As we will see below even groups like Z and R cannot be embedded into compact groups. We examine the question which topological groups admit representations on good Banach spaces (like: Hilbert, reflexive, ...). For these purposes we give a necessary basic material for topological . Including among others: first steps in uniform structures and uniformly continuous functions on groups. We touch also some questions from the van-Kampen Pontryagin theory for locally compact abelian groups. 6

2.1. First homework. Let G be a topological group. Exercise 2.6. Prove that (1) cl(A−1) = cl(A)−1 and cl(A)cl(B) ⊂ cl(AB) for every subsets A, B of G. (2) If H ≤ G is a then cl(H) ≤ G is also a subgroup. (3) If H ¢ G is a then cl(H) ¢ G is also a normal subgroup. (4) If G, in addition, is abelian and H ≤ G then cl(H) ≤ G is also an abelian subgroup. Give a counterexample if G is not Hausdorff. Exercise 2.7. Prove that the function

n k1 k2 kn (1) G → G, (x1, x2, ··· , xn) 7→ x1 x2 ··· xn is continuous for every given tuple n (k1, k2, ··· , kn) ∈ Z . (2) For every nbd U ∈ N(e) of the identity e ∈ G and every given natural n ∈ N there exists V ∈ N(e) such that V = V −1 and V n := VV ··· V ⊂ U. | {z } n times Exercise 2.8. Prove that (1) the inversion map, left and right translations, all conjugations, are homeomor- phisms G → G. (2) G is homogeneous as a topological space. (3) * For every pair (x, y) ∈ G × G there exists f ∈ Homeo(G, G) such that f(x) = y and f(y) = x. (4) Which of the following topological spaces are of the group type: (a) (R, τs) the Sorgenfrey line. (b) X := {x ∈ R2 : ||x|| = 5}. (c) X := {x ∈ R3 : ||x|| < 5}. (d) The integers Z with the cofinite topology. Exercise 2.9. Let A and B are subsets of G and g ∈ G. Prove that: (1) If A is open then gA and AB are open in G. (2) If A and B are compact then AB is also compact. (3) If A and B are connected then AB is also connected. (4) If A and B are closed then AB need not be closed. (5) * If A is closed and B is compact then AB is closed. (6) cl(A) = ∩V ∈N(e)VA = ∩V ∈N(e)VA. Exercise 2.10. Let G be a countable topological group which is either: a) metrizable by a complete metric; or b) locally compact and Hausdorff. Show that G is discrete. Hint: Use the Baire Category theorem: For every Hausdorff space X which is is either: a) metrizable by a complete metric; or b) locally compact the following holds.

For every countable cover X = ∪n∈NAn where each is An is closed in X at least one of the sets contains an interior point. That is, there exists k ∈ N such that int(Ak) 6= ∅. Remark: conclude that there is no complete metric on the space Q of all rationals.

Exercise 2.11. (1) Let {Gn}n∈N be a sequence of topological groups where each Gn is a (separable) metrizable topological group. Show that the topological product Q G endowed with the usual Tychonoff topology is a (separable) n∈N n metrizable topological group. 7

(2) ** Let {Gn}n∈N be a sequence of topological groups where each Gn is the topological group Q of all rational numbers carrying the usual topology. Let

G := ⊕n∈NGn = {x = (x1, x2, ··· ) : almost all coordinates are 0} be the direct sum endowed with the box topology τ inherited from Q G . box n∈N n Prove that (G, τbox) is a countable non-metrizable Hausdorff topological group. 8

3. Lecture 3 Proposition 3.1. (Basic properties of nbd’s at e) For every topological group G and every local base γ at e we have:

(1) ∀U1,U2 ∈ γ ∃V ∈ γ : V ⊂ U1 ∩ U2; (2) ∀U ∈ γ ∃V ∈ γ : V 2 ⊂ U; (3) ∀U ∈ γ ∃V ∈ γ : V −1 ⊂ U; (4) ∀U ∈ γ ∀a ∈ G ∃V ∈ γ : aV a−1 ⊂ U. Exercise 3.2. ∀G ∈ T Gr we have (1) ∀U ∈ N(e) ∀k ∈ N ∃V ∈ N(e): V −1 = V,V is open and V k ⊂ U. (2) ∀U ∈ N(e) ∀compact subset K ⊂ G ∃V ∈ N(e): xV x−1 ⊂ U ∀x ∈ K. Lemma 3.3. (Some useful properties) Let G be a topological group. Then (1) N(x) = xN(e) := {xU : U ∈ N(e)} and N(x) = N(e)x for every x ∈ G. For every local base γ at e the system xγ is a local base at x ∈ G. (2) G is discrete iff G contains an . (3) Every conjugation is a homeomorphism. (4) N(e)−1 = N(e). (5) for every O ∈ τ and every A ⊂ G we have AO and OA are open in G. (6) cl(A) ⊂ AV ∀V ∈ N(e) ∀A ⊂ G. (7) cl(A) = ∩{AV : V ∈ N(e)} = ∩{VA : V ∈ N(e)}. (8) If G is T2 then the Z(G) is closed in G. Proof. We show (6) cl(A) ⊂ AV ∀V ∈ N(e) ∀A ⊂ G. Let x ∈ cl(A). Then for every V ∈ N(e) we get xV −1 ∩ A 6= ∅ So, x ∈ AV. −1 (8) Using G ∈ T2 show that every stationary subgroup Sta := {x ∈ G : axa = x} −1 is closed. Indeed, Sta is the closed subset fa (e), where fa is the following continuous map −1 −1 fa : G → G, x 7→ axa x . Now observe that Z(G) = ∩{Sta : a ∈ G}.  3.1. Separation axioms. Theorem 3.4. Let G be a topological group. TFAE:

(1) G is T0. (2) G is T1 ({e} is closed in G). (3) G is T2 (Hausdorff). (4) G is T3 (regular). (5) * G is T3.5 (Tychonoff = completely regular). Proof. Here we prove only the equivalence of (1),(2),(3) and (4). (1) ⇒ (2): Let x 6= y ∈ G. Then by (1), without restriction of generality, say for x, there exists ∃ U ∈ N(x) s.t. y∈ / U 9

Then ∃V ∈ N(e): xV ⊂ U. From here x∈ / yV −1 but yV −1 ∈ N(y). (2) ⇒ (4): Well known Lemma: X ∈ T3 is equivalent to the following: for every x ∈ X and every U ∈ N(x) there exists V ∈ N(X) such that cl(V ) ⊂ U. By the homogeneity of G it suffices to verify this for x := e ∈ G. By Lemma 3.3 we have cl(A) ⊂ AV ∀V ∈ N(e) ∀A ⊂ G Let U ∈ N(e). Choose V ∈ N(e) s.t. V 2 ⊂ U. Then cl(V ) ⊂ V 2 ⊂ U. (4) ⇒ (1): Is trivial.  Corollary 3.5. G is Hausdorff if and only iff {e} is closed iff for every e 6= a ∈ G there exists U ∈ N(e): a∈ / U iff ∩{U : U ∈ N(e)} = {e}. Proposition 3.6. (1) Every open subgroup H ≤ G is clopen in G. (2) A subgroup H ≤ G is open iff int(H) 6= ∅. −1 n (3) Let U ∈ N(e) be symmetric (that is, U = U). Then H := ∪n∈NU is an open subgroup of G. Proof. (1) All cosets xH x ∈ G are open. So O := ∪{xH : x ∈ G, x 6= e} is open, too. Therefore, its complement G \ O = H is closed. (2) If O is a nonempty subset of G and if O ⊂ H then H = ∪{hO : h ∈ H}. −1 (3) Observe that HH ⊂ H and H = H. 

Theorem 3.7. Let G ∈ T Gr ∩ T2 be a Hausdorff topological group and H ≤ G be its topological subgroup. If H is locally compact then H is closed in G. Proof. It is equivalent to prove in the case of cl(H) = G. So we have to show that H is closed in cl(H). By Proposition 3.6 it suffices to show that H is open in G = Cl(H). Since H is LC one may choose a compact nbd K of e in H.

∃U ∈ NG(e) ∩ τ : U ∩ H ⊂ K

U = U ∩ G = U ∩ cl(H) ⊂ cl(U ∩ H) ⊂ cl(K) = K (remark1: for every open O ⊂ X and A ⊂ X we have O ∩ cl(A) ⊂ cl(O ∩ A)) (remark2: every compact subset is closed in a Hausdorff space)

So, U ⊂ K. Therefore, U ⊂ H. Hence, intG(H) 6= ∅. By Proposition 3.6 we conclude that H is open in cl(H). Hence, also closed. So, H = cl(H).  Corollary 3.8. It is impossible to embed a locally compact noncompact group into any Hausdorff . In particular, there is no finite-dimensional topologically faithful representation by linear isometries of a locally compact noncompact groups (like Z, R) on finite-dimensional Euclidean spaces. 10

Example 3.9. Show that every locally compact Hausdorff group G can be embedded into a compact Hausdorff semitopological monoid. Hint: Use the 1-point (Alexandrov) compactification. Definition 3.10. Let X be a topological space. A compactification of X is a contin- uous map f : X → Y where Y is a compact Hausdorff space and f(X) is dense in Y . We say: proper compactification when, in addition, f is required to be a topological embedding. One of the standard examples of a proper compactification is the so-called 1-point compactification ν : X,→ X∞ := X ∪ {∞} defined for every locally compact non- compact Hausdorff space (X, τ). Recall the topology

τ∞ := τ ∪ {X∞ \ K : K is compact in X}. Important example of a compactification is the so-called maximal (or, Stone-Chech) compactification β : X → βX which is proper iff X ∈ T3.5. See for example the file of Doron Ben Hadar downloadable from the course homepage. 11

3.2. Homework 2.

Exercise 3.11. Let G1, be topological groups and f : G1 → G2 be a homomor- phisms which is continuous at the point e ∈ G1. Show that f is continuous. Exercise 3.12. Let G be a topological group. Prove that: (1) ∀U ∈ N(e) ∀ compact subset K ⊂ G ∃V ∈ N(e): xV x−1 ⊂ U ∀x ∈ K; (2) for every compact subset K ⊂ G and a closed subset A ⊂ G with K ∩ A = ∅ there exists U ∈ N(e) s.t. UK ∩ A = ∅. Exercise 3.13. Show that for every connected topological group G and every nbd U ∈ N(e) we have n G = ∪n∈NU . Conclude that, in particular, U algebraically generates G. A topological group G is said to be compactly generated if there exists a compact subset K ⊂ G which algebraically generates G; that is < K >= G. For example, every compact group, Rn, Rn × TS. A topological space X is σ-compact if X = ∪n∈NKn where each Kn ⊂ X is compact. Exercise 3.14. (1) Show that every connected LC topological group G is com- pactly generated and σ-compact. (2) Give an example of a σ-compact topological group which is not compactly generated. Definition 3.15. Let (Y, τ) be a topological space and X be a set. Denote by Y X the set of all maps f : X → Y endowed with the product topology of Y X . This topology has the standard base α which consists of all the sets:

X O(x1, ··· , xn; U1, ··· ,Un) := {f ∈ Y : f(xi) ∈ Ui} where, F := {x1, ··· , xn} is a finite subset of X (all xi’s are pairwise distinct) and Ui are nonempty open subsets in Y . Other names of this topology are: pointwise topology, point-open topology. Exercise 3.16. (1) For every topological space X consider the semigroup (XX , ◦) of all selfmaps f : X → X wrt pointwise (=product) topology. Show that XX is a right topological semigroup. (2) C(X,X) is a semitopological subsemigroup of XX . * Is it true that C([0, 1], [0, 1]) is a topological semigroup ? X X (3) ** Prove that the left translation lf : X → X is continuous if and only if f ∈ C(X,X). Derive that if X is T1, then the right topological semigroup XX is semitopological iff X is discrete. Definition 3.17. Let X be a topological space. A compactification of X is a contin- uous map f : X → Y where Y is a compact Hausdorff space and f(X) is dense in Y . We say: proper compactification when, in addition, f is required to be a topological embedding. 12

One of the standard examples of a proper compactification is the so-called 1-point compactification ν : X,→ X∞ := X ∪ {∞} defined for every locally compact non- compact Hausdorff space (X, τ). Recall the topology

τ∞ := τ ∪ {X∞ \ K : K is compact in X}. Important example of a compactification is the so-called maximal (or, Stone-Chech) compactification β : X → βX which is proper iff X ∈ T3.5. See for example the file of Doron Ben Hadar downloadable from the course homepage. Exercise 3.18. (1) Let S := R ∪ {∞} be the 1-point compactification of R. Define the ”usual” operation + on S: x + y is already defined for x, y ∈ R. Otherwise, x + y = ∞ (that is, x+∞ = ∞+x = ∞+∞ = ∞). Show that (S, +) is a semitopological but not topological semigroup. (2) More generally, let (G, ·, τ) be a locally compact non-compact Hausdorff topo- logical group. Denote by S := G ∪ {∞} the 1-point compactification of G. * Show that (S, ·, τ∞) is a semitopological but not topological semigroup. Remark 3.19. As we know a locally compact Hausdorff group G admits an embedding into a compact Hausdorff group iff G is compact. Exercise 3.18 shows that such G at least admits a proper semigroup compactification ν : G,→ S such that S is a compact semitopological monoid. 13

4. Lecture 4 After Corollary 3.8 and Example 3.18 we give some remarks. The semitopological version of Example 3.18 is false as it follows from the following. Proposition 4.1. If S is a compact Hausdorff topological semigroup and if G is a subgroup of S then cl(G) is a (compact) topological group (note that if S is a monoid with the identity element eS then G is not necessarily a submonoid of S. That is, the identity element eG of G is not necessarily eS). Proof. (EXERCISE)  In particular, it follows that R cannot be embedded into the compact topological semigroup Θ(Rn) := {f ∈ L(Rn, Rn): ||f|| ≤ 1} of all non-expanding linear self- operators. It can be identified with the monoid of all matrices A = (aij) (size n × n) such that |aij| ≤ 1. Definition 4.2. Let G be a Hausdorff topological group. We say that f : G → S is a semigroup compactification of G if: (1) f is a compactification (see Definition 3.10); (2) S is a compact Hausdorff right topological semigroup (see Definition 1.4); (3) f is a homomorphism of semigroups; (4) f(G) ⊂ Λ(S), where Λ(S) := {a ∈ S : la : S → S is continuous} (topological centre of S).

Observe that then S is necessarily a monoid and f(eG) is its neutral element. Remark 4.3. (1) A Hausdorff topological group can be embedded into a compact Hausdorff topological semigroup iff G can be embedded into a compact Haus- dorff topological group. (2) Every LC T2 topological group G admits a proper semitopological compact- ification (as we already know a LC T2 topological group G admits a proper compact group compactification iff G admits a proper compactification into a compact topological semigroup iff G is compact). (3) * (MM, 2001) There exists a separable metrizable (complete) topological group G such that it does not admit a proper semitopological semigroup compact- ification. Namely, one may take G := H+[0, 1], the group of all orientation preserving homeomorphisms of the unit interval [0, 1] endowed with the com- pact open topology. The same topology on H+ can be defined by the following metric d(f1, f2) := sup |(f1(t) − f2(t)|) 0≤t≤1

The same group H+[0, 1] cannot be embedded into Isolin V for any reflexive (see Remark 4.4) Banach space V . (4) * An example of an important right topological semigroup compactification (which is not semitopological) is the maximal (Stone-Chech) compactification β : Z ,→ βZ of the group Z. Note that the standard semigroup structure on the semigroup βZ is not commutative. This compactification comes from the algebra Cb(Z) = l∞(Z) of all (continuous) bounded functions on Z. 14

(5) Any T2 topological group G admits a proper right topological semigroup compactification. Namely, one may consider the compactification βG : G,→ βGG. This compactification comes from the algebra RUCb(G) of all bounded right uniformly continuous functions f : G → R. The latter means that

∀ε > 0 ∀g0 ∈ G ∃U ∈ N(g0): |f(gx) − f(g0x)| < ε ∀g ∈ U ∀x ∈ G Remark 4.4. Recall that for every Banach space V one may define the dual Banach space V ∗ := {f : V → R} of all continuous linear functionals. Consider the canonical bilinear map V ∗ × V → R, (f, v) 7→ f(v). Then it naturally induces the canonical isometric inclusion map i : V → V ∗∗ into the second dual. When this map is onto then V is said to be reflexive. For example, every Hilbert space is reflexive. An example of a reflexive space which is not Hilbert is any lp with 1 < p < ∞, p 6= 2. A nonreflexive separable Banach space (with separable dual) is for example c0. 4.1. Homomorphisms and factor groups. Let q : X → Y be an onto map and (X, τ) ∈ TOP . Recall that the quotient topology on Y is defined by τY := {A ⊂ −1 Y : q (A) ∈ τ}. Then q is continuous and τY is the strongest topology making q continuous. If Y carries the quotient topology wrt q : X → Y then q is said to be a quotient map. Moreover we have the following useful lemma.

Lemma 4.5. Let f1 : X → Z, f2 : Z → Y be continuous maps. Suppose that f2 ◦ f1 : X → Y is an onto quotient map. Then f2 is also a quotient map.

Remark 4.6. In particular, if f2 is an identity continuous map then it is a homeomor- phism iff it is a quotient. A trivial example of a continuous onto homomorphism of topological groups which is not a quotient is the following map

id :(R, τdiscr) → R. Recall also that particular examples of quotient maps are closed continuous onto maps and open continuous onto maps. It turns out that for continuous onto homomorphisms between topological groups every quotient map necessarily is open (not always closed, however; take for example any projection R2 → R). Proposition 4.7. Let H ¢ G ∈ T Gr. Then: G/H wrt quotient topology is a topo- logical group and the canonical onto homomorphism q : G → G/H, x 7→ [x] = xH is continuous and open. Proof. The continuity of q follows from the definition of the quotient topology. We show that q is open. Observe that q−1(q(U)) = UH = ∪{uH : u ∈ U} = ∪{Uh : h ∈ H} is open for every open U ⊂ G as a union of open sets. Now it is easy to see by definitions that G/H is a topological group.  Exercise 4.8. R/Z is topologically isomorphic to T. More generally, Rn/Zn is topo- logically isomorphic to Tn. 15

Proof. Observe that f : R → T, f(t) := cis(2πt) = e2πit is open because it is ”locally open”. Indeed, for every x0 ∈ R the f(x0 − ε, x0 + ε) is open in T for every 0 < ε ≤ 1. Now use Lemma 4.5 and Remark 4.6.  Lemma 4.9. (Properties of the quotient groups G/H) (1) G/H is Hausdorff iff H is closed in G. (2) G/H is discrete iff H is open in G. (3) Any closed subgroup H ≤ G of the finite index [G : H] < ∞ is clopen. (4) If G is LC then G/H is also LC. (5) (up to the Kakutani thm mentioned above) If G is metrizable and H is closed then G/H is also metrizable (=B1). (6) If f : G → Y is a continuous homomorphism and H ⊂ kerf then there exists a unique continuous homomorphism ν : G/H → L such that f = ν ◦ π, where π : G → G/H is the natural map. (7) Let f : K → H be a continuous onto homomorphism of a compact group onto the Hausdorff group H. Prove that f is open. (8) R/Z is isomorphic to T in TGr. Proof. Exercise.  Lemma 4.10. (Hausdorff) quotients of G (up to an isomorphism in TGr) are the quotient groups G/H where H is a (closed) normal subgroup of G.

Proof. Use Lemma 4.5 and the first isomorphism theorem for abstract groups.  16

4.2. Homework 3. Exercise 4.11. * Prove or disprove: the topological group T2 (two-dimensional torus) is a topological factor group of the group C r {0}.

Exercise 4.12. Prove or disprove: GLn(R)/D is a locally compact Hausdorff topolog- ical group, where D denotes the set of all invertible scalar matrices in GLn(R) for every n ∈ N.

Exercise 4.13. Give a concrete example of a continuous onto homomorphism f : G1 → G2 of Hausdorff separable metrizable topological groups which is not a quotient map. Exercise 4.14. Let G be a topological group and H be its normal subgroup. Prove: (1) G/H is discrete iff H is open in G. (2) If G is Hausdorf and the normal subgroup H ≤ G is closed and has the finite index [G : H] < ∞ then H is clopen. (3) If G is LC then G/H is also LC. (4) If G is metrizable then G/H is also metrizable (Hint: You may use the Kaku- tani thm mentioned above). (5) Let f : G1 → G2 be a continuous homomorphism of topological groups. As- sume that G1 is compact and G2 is Hausdorff. Show that f is an open map. (6) If f : G → Y is a continuous homomorphism of topological groups and H E G, H ⊂ kerf then there exists a unique continuous homomorphism ν : G/H → L such that f = ν ◦ π, where π : G → G/H is the natural map. Exercise 4.15. * If S is a compact Hausdorff topological semigroup and if G is a subgroup of S then cl(G) is a (compact) topological group. Hint: eG is an idempotent of S and also an identity of T := cl(G). 17

5. Lecture 5 5.1. Closed subgroups of R and T, Kronecker’s theorem. First of all we will examine the following questions: what are the closed subgroups of R and T ? what are the dense subgroups ? We are going to prove that: the closed subgroups of R are only: R, aZ a ≥ 0 Every nondiscrete subgroup H ≤ R is dense in R. the closed subgroups of T are only:

T, Ωn n ∈ N Lemma 5.1. For every a ∈ R the set < a >= aZ is a closed discrete (and, of course, cyclic) subgroup of R. Theorem 5.2. Every nondiscrete subgroup H ≤ R is dense in R. Proof. Assume that H is not discrete. Equivalently, 0 is not isolated in H. Hence, ε ε ∀ ε > 0 ∃ g ∈ (− , ) ∩ H 6= ∅ ε 2 2 Clearly,

Mε :=< gε >:= {kgε : k ∈ Z} ≤ H.

This subgroup Mε is ε-dense in R. Then also H is ε-dense in R. It follows that H is topologically dense in R (being ε-dense for every ε > 0).  Theorem 5.3. The closed subgroups of R are only: R, aZ (a ≥ 0) Proof. Assume that H is a closed subgroup of R and H 6= R. Then H is not dense in R. Therefore, by Theorem 5.2 H is necessarily discrete. If H = {0} then take a = 0. Without restriction of generality suppose that H 6= {0} is a discrete proper sub- group. Then there exists b ∈ H ∩ (0, ∞). The subset H ∩ [0, b] is compact (being closed and bounded in R). Also, H ∩ [0, b] is discrete because H is discrete. It follows that H ∩ [0, b] is finite. So, there exists the smallest positive element in H ∩ [0, b]. Denote it by a. We show that H =< a >. Indeed, for every x ∈ H we have: x (1) x − [ a ]a ∈ H; x (2) 0 ≤ x − [ a ]a < a. x x Recall the following property (of the fractional part) 0 ≤ a −[ a ] < 1. This explains the second fact. The first fact is easy because H is a subgroup and the part x [ a ] ∈ Z. x Now by the minimality of a we necessarily have 0 = x − [ a ]a. So, x ∈ aZ.  Theorem 5.4. The list of all (up to a topological isomorphism) Hausdorff quotients of R is: R, {0} and T. 18

Proof. Only the case of T is nontrivial. By Lemma 4.10 every proper nontrivial Hausdorff quotient of R is R/aZ for some nonzero a ∈ R. Observe that R is a topological field. Hence, for every nonzero a ∈ R the map fa : R → R, x 7→ ax is an automorphism of the topological group (R, +). Then it is clear that all R/aZ (with a 6= 0) are naturally isomorphic. Finally recall that R/Z is isomorphic to T in TGr.  Theorem 5.5. Let a, b ∈ R. Consider the subgroup H :=< a, b >= {na + mb : n, m ∈ Z} ≤ R Then H is dense in R iff a, b are rationally independent. Proof. By Theorem 5.3 we can conclude that H is not dense in R iff H is discrete and cyclic. In this case there exists c ∈ R such that a, b ∈ cZ. So, a = k1c, b = k2c for a b some k1, k2 ∈ . Then c = = (WRG we can assume that a and b are nonzero). Z k1 k2 So, a, b are rationally dependent, a contradiction.  Example 5.6. (1) The subgroup √ √ H =< 1, 2 >= {m + n 2 : m, n ∈ Z} is dense in R. √ (2) The subgroup A =< cis2π 2 > is dense in T. Exercise 5.7. (Cyclic subgroups of T) Let a := cis(2πα) ∈ T. Then the O(a) is infinite if and only if α is irrational.

Let Ω∞ := ∪m∈NΩm. Then: (1) Ω∞ = {cis(2πα): α ∈ Q}. (2) The group Ω∞ is isomorphic to Q/Z. (3) Ω∞ is a dense subgroup of T. Theorem 5.8. (Kronecker’s theorem for dimension 1, ”Irrational Billiard” in the circle group) For every irrational α the cyclic subgroup H :=< cis(2πα) > is dense in T. Proof. For every irrational α ∈ R the (necessarily noncyclic) subgroup < 1, α >< R is dense in R. Indeed, otherwise, its closure cl(K) is a closed proper subgroup of R such that cl(K) is not cyclic. It is a contradiction by Theorem 5.3. So, K is dense in R. Then q(K) is dense in T, where q : R → T is the natural continuous s onto homomorphism. Now observe that q(K) = H =< cis(2πα).  A topological group G is said to be monothetic if it contains a cyclic dense subgroup. R is not monothetic because any cyclic subgroup of R is discrete and closed. Corollary 5.9. Every infinite cyclic subgroup of T is dense in T. In particular, T is monothetic.

Note that Tn is also monothetic for every natural power. 19

Theorem 5.10. Every closed proper subgroup H of T is the finite m Ωm := {z ∈ C : z = 1} for some m ∈ Z. Proof. q−1(H) ≤ R is closed and proper. By Theorem 5.3 we know that q−1(H) = aZ is cyclic for some a ∈ R. Then its image H is also cyclic. By our assumption the cyclic subgroup H is closed. Therefore, by Corollary 5.9 and Exercise 5.17 we conclude that H is Ωm for some m ∈ N.  Corollary 5.11. The Hausdorff quotients of T are T and {0} (up to the topological isomorphisms). Moreover, for every quotient onto homomorphism γ : T → T there exists m ∈ Z such that γ(z) = zm ∀z ∈ T. Proof. By Lemma 4.10 we need to classify the quotients T/H, where H is a closed subgroup. By Theorem 5.10 we have only to classify the quotients T/Ωm, where m ∈ Z. It is enough to show that T/Ωm is isomorphic to T in TGr for every m ∈ N. This follows from the following commutative diagram

ϕm / T I T II O II II γ p II I$ T/kerϕm m taking into account that ϕm : T → T, z 7→ z is an onto continuous homomorphism for every m ∈ N and kerϕm = Ωm. Since ϕm : T → T is a continuous closed (compactness of the Hausdorff group T) onto map, it should be a quotient. Then it follows by Lemma 4.5 that γ is a topological homeomorphism (hence, an isomorphism in TGr).  Remark 5.12. One may reformulate Theorem 5.8 (1-dimensional Kronecker’s approx- imation thm) as follows. For every irrational β the sequence nβ − [nβ](n ∈ Z) is dense in [0, 1). Use the fact that the map q : R → R/Z is of period 1. Equivalent formulations of Theorem 5.8: (1) For every ε > 0 and every r ∈ R there exists an integer m ∈ Z with |mα − r| < ε (mod 1) (2) For every ε > 0 and every a ∈ R there exists an integer m ∈ Z with |mα − r − n| < ε ∃n ∈ Z Theorem 5.13. (Kronecker’s approximation theorem) Let 1, α1, α2, ··· , αn be a rationally independent finite family of real numbers. Then for every ε > 0 and every n-tuple (r1, r2, ··· , rn) of real numbers there exists an integer m ∈ Z with |mαk − rk| < ε (mod 1) ∀k ∈ {1, ··· , n} (equivalently, |mαk − rk − nk| < ε ∀k ∈ {1, ··· , n} ∃nk ∈ Z) Corollary 5.14. Every Tn is monothetic. That is, there exists a cyclic dense sub- group. 20

5.2. Homework 4. Exercise 5.15. Let G be a Hausdorff nondiscrete topological group. Show that in the topological group G2 there exist at least 3 distinct but isomorphic proper (6= G2) nondiscrete closed subgroups. Exercise 5.16. Show that there exist: an abelian topological group (G, +) and closed subgroups H1 and H2 of G, such that the subgroup H1 + H2 is not closed. Show also that it is impossible for G := T. Exercise 5.17. (Cyclic subgroups of T) Let a := cis(2πα) ∈ T. Then the order O(a) is infinite if and only if α is irrational.

Let Ω∞ := ∪m∈NΩm. Then: (1) Ω∞ = {cis(2πα): α ∈ Q} ≤ T. (2) The group Ω∞ is isomorphic to Q/Z. (3) Ω∞ is a dense subgroup of T. Exercise 5.18. Let G be a Hausdorff monothetic group. (1) Show that G is abelian. (2) * Give an example of a monothetic group G having a non-monothetic sub- group. (3) If G is metrizable then the cardinality of the set G is not greater than the cardinality of R. Definition 5.19. A Hausdorff topological group G is said to be minimal if there is no strictly coarser Hausdorff group topology on G. Exercise 5.20. * Which of the following topological groups are minimal: Z, R, T ? Exercise 5.21. Let S be the interval [0, 1] with the multiplication

( 1 t, if 0 ≤ t < 2 ; st = 1 1, if 2 ≤ t ≤ 1. Show that: S is a compact right topological semigroup with Λ(S) = ∅. The subset 1 1 T := [0, 2 ) is a subsemigroup of S and cl(T ) = [0, 2 ] is not a subsemigroup of S. Exercise 5.22. Let S := Z∪{−∞, ∞} be the two-point compactification of Z. Extend the usual addition by: n + t = t + n = s + t = t n ∈ Z, s, t ∈ {−∞, ∞} Show: (S, +) is a noncommutative compact right topological monoid having dense commutative topological centre Λ(S) = Z. S is not semitopological. 21

6. Lectures 6 and 7 Theorem 6.1. (1) The function 2 f : R → T , x 7→ ([x], [αx]) = (cis(2πx), cis(α2πx)) is a continuous not onto homomorphism with the dense image. (2) The subgroup H := {([x], [αx]) : x ∈ R} < T2 is dense in T2. (3) The image of the line {(x, y) ∈ R2 : y = αx} into the torus T2 is dense. Proof. We have a flow (a of R on T2) R × T2 → T2 defined by

t([x1], [x2]) := ([x1 + t], [x2 + αt]) We show that the Z-orbit (hence, also R-orbit) of the point ([0], [0]) ∈ T2 is dense in T2. It suffices to check that this orbit intersects every ”meridian circle”

Sa := {([a], [x]) : x ∈ R} in a dense subset of Sa for every given a ∈ R. Claim: the point ([t], [αt]) of the orbit of ([0], [0]) is in Sa iff t ∈ a + Z. Hence the set of all second coordinates of such points is

{[α(a + n)] : n ∈ Z} = {[αa + αn]: n ∈ Z} = [αa] + H0 where H0 := {[αn]: n ∈ Z}. But H0 = q(αZ) is dense in T because α is irrational (Theorem 5.8, Kronecker’s theorem for dimension 1). Then its translation [αa] + H0 is also dense in T. 

All topological groups below are assumed to be Hausdorff. Theorem 6.2. (Hewitt and Zuckerman) Let G ∈ LCA. Then the set of all con- tinuous characters pointwise approximates the set of all characters. Precisely, let f : G → T be a (not necessarily continuous) character. Then for every ε > 0 and every finite subset F ⊂ G there exists a continuous character χ : G → T such that |f(x) − χ(x)| < ε for every x ∈ F .

Theorem 6.3. Let 1, α1, α2, ··· , αn be a rationally independent finite family of real numbers. Then for every ε > 0 and every n-tuple (r1, r2, ··· , rn) of real numbers there exists an integer m ∈ Z with

|mαk − rk| < ε (mod 1) ∀k ∈ {1, ··· , n}

(equivalently, |mαk − rk − nk| < ε ∀k ∈ {1, ··· , n} ∃nk ∈ Z)

Proof. Treat (R, τdiscr) as an (infinite dimensional, of course) vector space over the field Q. Consider the finite dimensional Q-linear subspace H := span{α1, ··· , αn} < R. Then α1, ··· , αn is its linear basis because this family is Q-independent. WRG we can suppose that 1 ∈ H (otherwise, if 1 ∈/ H then the family

α0 := 1, α1, α2, ··· , αn is Q-independent and we can continue with this family and add arbitrarily r0 ∈ R). 22

P n So we can suppose that 1 = ciαi with some tuple (c1, ··· , cn) ∈ Q . There exists a unique Q-linear functional λH : H → R such that

fk(αk) = rk, ∀k ∈ {1, ··· , n}. Since R is a divisible group there exists a homomorphic extension (not necessarily continuous wrt the standard topology of R) λ :(R, τdiscr) → R. Now we define the P desired character f : R/Z → R/Z. Let a := ciri ∈ R. Consider two cases: (1) if a 6= 0 then consider

2πix 2π iλ(x) f : T → T, f(e ) = e a We obtain that f is a (possibly, discontinuous) character of T. Observe that it is well defined because λ(1) = a, hence λ(s) = sa for every s ∈ Z and we get 2πi(x+s) 2π iλ(x+s) 2π iλ(x)+ 2π iλ(s) 2π iλ(x) 2πix f(e ) = e a = e a a = e a = f(e ) ∀s ∈ Z (2) if a = 0 then define 2πix 2πiλ(x) f : T → T, f(e ) = e .

Apply Hewitt-Zuckerman theorem to this function. Then for every ε > 0 and every finite subset F ⊂ G there exists a continuous character χ : G → T such that |f(x) − χ(x)| < ε for every x ∈ F . On the other hand every continuous character χ ∈ T∗ has the m form um : T → T, um(t) = t for some m ∈ Z. Define F := {q(αk)} ⊂ T. Now substituting um(q(αk)) = um(x) instead of χ(x) and f(αk) = rk we eventually get: there exists an integer m ∈ Z with

|e2πimαk − e2πirk | < ε ∀k ∈ {1, ··· , n} or

|mαk − rk| < ε (mod 1) ∀k ∈ {1, ··· , n} Definition 6.4. Let G be a topological group. A (continuous) homomorphism f : G → T is called to be a (continuous) character of G. Denote by G∗ or Hom(G, T) the set of all continuous characters of G. It is a group wrt the following pointwise operation

(γ1 + γ2)(g) := γ1(g)γ2(g) This group G∗ wrt compact-open topology is a LCA topological group which is said to be the Pontryagin dual of G. Recall that the family [K,O] := {f : G → T : f(K) ⊂ O} with compact K ⊂ G and open O ⊂ T is a subbase of the compact open topology on G∗. (1) we have a contravariant functor LCA → LCA. (2) The natural evaluation map w : G × G∗ → T is continuous. It induces a map G → G∗∗. This map is continuous. 23

(3) Moreover, by Pontryagin-vanCampen duality theorem this map is a topologi- cal isomorphism for every LCA group G. An (D, ·) is said to be divisible if for every n ∈ N the canonical n endomorphism un : D → Dn, x 7→ x is onto. Examples: T, R, C but not Z.

Lemma 6.5. Let H be a subgroup of an abelian group G and f0 : H → D be a homo- morphism into a divisible group (D, ·). Then there exists a homomorphic extension f : G → D of f0. By Zorn’s Lemma it is enough to show that for every x∈ / H there exists a homo- k morphic extension on the subgroup H0 :=< x, H >= {x h : k ∈ Z, h ∈ H}. n n Case (i): If x 6= H ∀n ∈ N then define f(x h) = f0(h). k k Case (ii): Let k ≥ 2 is the least natural number s.t. x ∈ H. Let f0(x ) = d ∈ D. Since D is divisible one may choose z ∈ D s.t. zk = d. Now define f(xnh) = n z f0(h).  Corollary 6.6. Let G be an abelian group. Then the characters into T separate the points. Proof. Let g 6= e. It is enough to show that there exists χ ∈ G∗ s.t. χ(g) 6= 1 = e ∈ T. Case (1). Let O(g) = n. Then consider the embedding f0 : H :=< g >,→ Ωn < T. Case (2). Let (O(g) = ∞. Then consider any homomorphism f0 : H :=< g >→< z >< T with z = f0(g) 6= 1.  Theorem 6.7. (1) Z∗ = T. m For every m ∈ Z consider χm : T → T, x 7→ x . w : Z → T → T n Consider χt : Z → T, n 7→ t (where t ∈ T given constant). Observe that every homomorphism χ : Z → T has this form for some t ∈ T. Show that Z∗ is algebraically (in fact, even, topologically) isomorphic to T. (2) T∗ = Z. Z × T → T, (k, t) 7→ tk. (3) R∗ = R. 2πirx χr : R → T, x 7→ e .

2πist w : R → R → T, (s, t) 7→ e Zn × Tn → T and Rn × Rn → T are well defined biadditive mappings. In general: G × G∗ → T is a continuous biadditive mapping for every LCA group G. The dual operator preserves the finite products. That is, n n Y ∗ ∼ Y ∗ ( Gi) = Gi i=1 i=1 LCA group G is said to be autodual if it is t. isomorphic to its dual G∗. Exercise 6.8. Show that every LCA group G is a subgroup of an autodual LCA group H.

Proposition 6.9. Every character γ : T → T has the form γ = χm for some m ∈ Z. 24

Proof. Let K := Ker(γ). Then K is a finite cyclic group Ωm or T. In the latter case we have the trivial character and we can take m = 0. So wrg suppose that K = Ωm for some m ∈ N. Then T/Ωm is topologically isomorphic to T. Moreover, there exists σ :  Theorem 6.10. Let G be a LCA group. (1) If G is discrete then G∗ is compact. (2) If G is compact then G∗ is discrete.

Proof. (1) If X is discrete then for C(X,Y ) we have τc−o = τp. G∗ ⊂ Hom(G, T) ⊂ TG ∈ Comp. G∗ is pointwise closed in TG. (2) If X is compact then for C(X,Y ) we have τc−o = τsup. (G, U 1 ) ∈ N(e). 4 On the other hand (G, U 1 ) = {1} because 4

U 1 ⊃ χ(G) ≤ ⇒ χ(G) = 1 4 T So, the identity is an isolated point.  Remark 6.11. G∗ is a compact metrizable iff G is a countable abelian group. Indeed, in Thm 6.10.1 we have G∗ ⊂ TG ∈ Comp for every discrete abelian G. Hence, if G is countable then G∗ is metrizable being a subspace of the metrizable group TG. In fact, the converse is also true. Some categorical facts: (1) (Pontryagin duality) In order to study the theory of compact (metrizable) abelian groups it is equivalent to study the theory of discrete (countable) abeiian groups. The cat- egory of compact abelian (metrizable) groups is dual (meaning: is equivalent to the dual) of the category of abelian discrete (countable) groups. (2) (Gelfand duality) Category of all compact Hausdorff spaces is dual to the category of all commutative Banach algebras. (3) (Stone duality) Category of all 0-dimensional compact Hausdorff spaces is dual to the cat- egory of all Boolean algebras.

Compactly generated groups: Definition Examples: compact groups, Rn, n Lemma 6.12. For every symmetric nbd U ∈ N(e) the subset H := ∪nU is a clopen subgroup. Lemma 6.13. For every connected LC group G and any U ∈ N(e) we have G =< U >. Theorem 6.14. Every connected LC group is compactly generated. As a nontrivial generalization of the classification of finitely generated abelian groups Pontryagin duality provides the following 25

Remark 6.15. {compactly generated LCA groups} = = {Rn×Zm×K}, where K runs over compact abelian groups and n, m ∈ {0, 1, ···}. {compactly generated connected LCA groups} = {Rn × K} Theorem 6.16. On every abelian group G there exists a precompact Hausdorff group topology (which is metrizable if G is countable).

Proof. There exists a set S ⊂ Hom(G, T) which separates the points of S and Q S |S| = card(G) for infinite G. The diagonal function f : G → T is an injec- tive homomorphism into the compact group TS. So, f(G) is a precompact group. If G is countable then we can choose a countable S. Therefore, TS (hence, f(G)) is metrizable. Since f : G → f(G) algebraically is an isomorphism we can endow G with the ”preimage topology” coming from f(G).  Corollary 6.17. On every infinite abelian group G there exists a nondiscrete Haus- dorff group topology. Proof. Every discrete (even every LC) subgroup of a Hausdorff group is closed. Hence, if in Thm 6.16, f(G) would discrete in the COMPACT group TS then f(G) is finite (being discrete and compact).  Remark 6.18. It is not true for nonabelian groups (Hesse, independently, Olshanskii (and Shelah under assumption CH)). In other terms there exists an infinite which is minimal. That is an infinite group G which is miniamal, that is, does not admit a nondiscrete Hausdorff group topology. Recall Definition 6.19. A Hausdorff topological group G is said to be minimal if there is no strictly coarser Hausdorff group topology on G.

Let us explain why R is not minimal (one of the homework exercises). We use the following reformulation of the minimality. Fact. A Hausdorff topological group G is minimal if and only if for every con- tinuous injective onto homomorphism f : G → Y is a homeomorphism (or, in other words, if any morphism in TGr which is an isomorphism in the category Gr is an isomorphism in the category TGr). Consider the following injective continuous not onto homomorphism √ 2πit 2πi 2t f : R → T × T, f(t) = (e , e ) By a corollary of Kronecker’s theorem f(R) is dense in T × T. f is not a topological embedding. Otherwise, f(R) is a locally compact subgroup in T × T. By a theorem (we have proved earlier) we obtain that f(R) is closed in T × T. Since f(R) is dense we obtain f(R) = T × T, a contradiction. Below all topological groups are assumed to be Hausdorff. Remark 6.20. (Zoo of minimal groups) (1) Every compact group is minimal. (2) (Stephenson) Every LCA minimal group is necessarily compact. 26

(3) (Prodanov, Stoyanov) Every abelian minimal topological group is precom- pact. (Observe that one may easily derive Stephenson’s theorem (2) using this result). (4) (Doitchinov) (Z, τp) is minimal. (5) (Stoyanov) The Isolin(H) is minimal for every Hilbert space H. (6) (Dierolf and Schwanengel) The topological R n R+ is min- imal. (7) (Gaughan) The SX in the pointwise topology (for every set X). (8) (Remus and Stoyanov) SLn(R) for every n ≥ 2. (9) (Me) Every LCA group G is a group retract of a LC minimal group – semidirect product – (T × G∗) n G which in fact is the generalized Heisenberg group modelled on the biadditive mapping G × G∗ → T. (10) (Gamarnik) The homeomorphisms group (in the compact open topology) Homeo([0, 1]N) and Homeo({0, 1}N) are minimal. Note that [0, 1]N is the Hilbert cube and {0, 1}N is the Cantor cube (home- omorphic to the usual Cantor set). Theorem 6.21. Every compact abelian group is a subgroup of some (maybe infinite- dimensional) torus TS. Proof. As an exercise.  What about nonabelian case ? Peter-Weyl theorem asserts that for every compact group G the homomorphisms into finitely dimensional orthogonal groups separate the points. Conclude that every compact group ,→ product of orthogonal groups. 27

7. Lecture 8 Recall Proposition 7.1. For every topological group G and every local base γ at e we have:

(1) ∀U1,U2 ∈ γ ∃V ∈ γ : V ⊂ U1 ∩ U2; (2) ∀U ∈ γ ∃V ∈ γ : V 2 ⊂ U; (3) ∀U ∈ γ ∃V ∈ γ : V −1 ⊂ U; (4) ∀U ∈ γ ∀a ∈ G ∃V ∈ γ : aV a−1 ⊂ U.

(T2) G is Hausdorff if and only if ∩{U : U ∈ γ} = {e}. Theorem 7.2. (Group topologization) Let G be a group and γ be a nonempty family of subsets containing e and satisfying the conditions (1), (2), (3) of Proposition 7.1. Then there exists a unique topology τ on G such that: (G, τ) is a topological group and γ is a local topological base at e.

(T2) (G, τ) is Hausdorff if and only if ∩{U : U ∈ γ} = {e}. Sketch of the proof: Define the desired topology τ on G as follows: O ∈ τ iff ∀x ∈ O ∃U ∈ γ xU ⊂ O. Remark 7.3. Suppose that in Theorem 7.2, in addition, the following condition is satisfied: (*) ∀U ∈ γ ∀x ∈ U ∃V ∈ γ xV ⊂ U Then γ ⊂ τ (that is, every nbd U ∈ γ is open). Observe that if U is a subgroup of G then this condition is trivially holds (take V := U) and hence, U is open.

n Example 7.4. Let G := Z and γ := {p Z}n∈N with a given prime p. Then the conditions of γ Theorem 7.2 are satisfied. The corresponding group topology τp is n n Hausdorff because ∩np Z = {0} and every p Z is an open subgroup (hence, clopen) in Z. It is easy to see that τp = top(dp), where dp is the standard p-adic topology on Z.

Exercise 7.5. Show that the topologies τp 6= τq on Z are different for different p, q.

Remark 7.6. Let G be a group, (Gi, τi) is a family of topological groups and fi : G → Gi be a family of homomorphisms. Then the so-called weak topology τw on G is a group topology. It is the smallest topology on G such that all fi are continuous. One way to see that it is a group topology is to use Theorem 7.2. Indeed, define γ as the −1 fin fin collection {fi (Ne): i ∈ I} (where S means the family of all finite intersections of members from the family S). An important and well known particular case of the weak topology is the topological product defined for the collection of the projections. It is easy to interpret the p-adic topology τp as the weak topology Z with respect to the collection of homomorphisms into the discrete (cyclic finite) groups fn : Z → Zpn , −1 n n ∈ N. Observe that the subgroups fn (0) = kerfn = p Z consist the local base. Example 7.7. Let G is a group and γ is a collection of subsets in G such that: 28

(1) Every H ∈ γ is a normal subgroup of G. (2) H1 ∩ H2 ∈ γ for every H1,H2 ∈ γ. Then γ is satisfies the conditions of Theorem 7.2. Example 7.8. On every group G one may define the corresponding profinite topology as the weak topology with respect to all possible homomorphisms G → F , where F is a finite group. Another description is as follows. Consider γ as a collection of all normal subgroups with the finite index. γ := {H/G :[G : H] < ∞} Then γ satisfies the conditions of Theorem 7.2. Warning: the profinite topology is not always Hausdorff. Definition 7.9. A group G is said to be residually finite if the profinite topology on G is Hausdorff. It is equivalent to saying that homomorphisms into finite groups separate the points of G. Other equivalent form of this definitions is: the intersection of all normal subgroups with finite index is {e}. Moreover, one may say also: the intersection of all (not necessarily normal) subgroups with finite index is {e}. The reason is that if H is a finite index subgroup of G then there exists a finite index normal subgroup of G such that N ⊂ H. Indeed, if G/H is finite then consider the standard left action G×G/H ×G/H → G/H and the corresponding homomorphism h : G → SG/H . Now observe that the kernel of this action N := ker(h) is the desired subgroup. Some examples of residually finite groups: finite groups, free groups, free abelian groups, finitely generated linear groups, finitely generated nilpotent groups. There exist finitely generated not residually finite grouos. One may take for example a subgroup G =< a, b >≤ SZ of the symmetric group SZ, where G is generated by two elements a, b, where a(n) = n + 1 and b(0) = 1, b(1) = 0, b(n) = n for n∈ / {0, 1}. Example 7.10. (Furstenberg’s proof of the infiniteness of prime numbers using the profinite topology on Z) Let (Z, τpro−fin) be Z in its profinite topology. Clearly, the subset {1, −1} is not open (in fact, every open subset is infinite (because every nbd of 0 contains an infinite subgroup)). We have to prove that the set P of all primes is infinite. Assuming the contrary let P := {p1, ··· , pn}. Then n Z \{1, −1} = ∪i=1piZ Every piZ is an open subgroup of (Z, τpro−fin) (being the kernel of the homomorphism

Z → Zpi into the discrete group Zpi ). Then, as we already know, every open subgroup n is closed. So, the finite union ∪i=1piZ is closed, too. Then we obtain that its complement {1, −1} is open, a contradiction.

Example 7.11. (Symmetric groups) For every set X denote by SX the symmetric X group of all bijections X → X. Consider the pointwise topology τp on SX ⊂ X . Then SX is a Hausdorff topological group. One may verify this directly. Another way is to use Theorem 7.2. Indeed, define

γ := {HA : A ⊂ X,A is finite} HA := {f ∈ SX : f(a) = a ∀a ∈ A} 29

It is trivial to see the conditions (1), (2) and (T2) of Theorem 7.2. As to (3), observe that for every f ∈ SX and every finite A ⊂ X we have −1 fHf −1(A)f = HA. ∗ SX := {f ∈ SX : f(x) = x for almost all x ∈ X} is a dense normal subgroup of SX . Example 7.12. (I-adic topology on rings) Let R be a commutative unital ring and I ⊂ R be an ideal in R. Consider n γ := {I : n ∈ N} Then γ satisfies the conditions (1), (2), (3) of Theorem 7.2. The corresponding topology τI is a topology on R. n Clearly, O ∈ τI if and only if ∀x ∈ O ∃n ∈ N x + I ⊂ O. Definition 7.13. A topological group G is said to be non-archimedean if there exists a nbd base γ at e such that every U ∈ γ is a (necessarily, open, see Remark 7.3) subgroup of G. Notation G ∈ NA. Note that the class NA is important in many applications. NA is closed under formation of: subgroups, factor-groups, products, completions. NA contains the class of all profinite groups and also all symmetric groups SX . It is a remarkable fact that there is an NA analog of Cayley’s theorem. Theorem 7.14. Every Hausdorff NA group is embedded (as a topological subgroup) into some SX . Definition 7.15. A topological group G is said to be profinite if G is a closed sub- Q group of the product i Fi of some finite groups. It is equivalent to say that G is a zero-dimensional compact Hausdorff topological group, or, G is an inverse limit of finite groups. For example, the completion of (Z, dp) is a profinite group. 30

8. Lecture 9 1 Theorem 8.1. Let G be a topological group. Suppose that {Un}n∈N is a sequence 3 of symmetric nbds of e in G such that Un+1 ⊂ Un for every n ∈ N. Then there is a right invariant pseudometric ρ on G such that 1 1 (∗) B 1 (e) := {x ∈ G : ρ(e, x) < } ⊂ Un ⊂ B 1 [e] := {x ∈ G : ρ(e, x) ≤ } 2n+1 2n+1 2n 2n Proof. For every r > 0 define Wr ∈ N(e) as follows. Let Wr := G for every r ≥ 1. For every 0 < r < 1 define 1 1 Wr := ∪{Ui1 ··· Ui : + ··· + < r} k 2i1 2ik Observe that x ∈ Wr iff there are x0, x1, ··· , xk ∈ G such that x0 = 1, xk = x and −1 xj−1xj ∈ Uij . It is trivial to see that −1 Wr = Wr and WrWs ⊂ Wr+s for every r, s ∈ (0, 1]. Define −1 ρ(x, y) := inf{r ∈ (0, 1] : xy ∈ Wr} Then 1) ρ(x, x) = 0 (because, e ∈ Wr for every r). −1 2) ρ(x, y) = ρ(y, x) (because, Wr = Wr). 3) ρ(x, z) ≤ ρ(x, y) + ρ(y, z). −1 −1 −1 Because if ρ(x, y) < r, ρ(y, z) < s then xy ∈ Wr, yz ∈ Ws. Therefore, xz ∈ WrWs ⊂ Wr+s. So, ρ(x, z) < r + s. 4) ρ(xg, yg) = ρ(x, y) (because (xg)(yg−1) = xy−1). Lemma. If 1 + ··· + 1 < 1 then U ··· U ⊂ U . 2i1 2ik 2n+1 i1 ik n We use induction wrt k. 1 1 For k = 1. Let 2i1 < 2n+1 . Then i1 > n + 1. So, Ui1 ⊂ Un+1 ⊂ Un (because Un is a decreasing sequence). Suppose that the inclusion holds for every k ∈ {1, ··· , k − 1}. Now we prove the induction step for k. Let m be the largest natural number such that 1 1 1 + ··· + < . 2i1 2ik 2m Then we have il > m (trivial). Let l < k be the smallest natural number that satisfies 1 1 1 < + ··· + . 2m+1 2i1 2il Then m ≥ n + 1 (because the induction assumption 1 + ··· + 1 < 1 and the 2i1 2ik 2n+1 choice of m) and 1 1 1 + ··· + < 2i1 2il−1 2m+1 1 1 1 + ··· + < 2il+1 2ik 2m+1 1present proof is from [2] 31

Using the inductive hypothesis we obtain

Ui1 ··· Uil−1 ⊂ Um and

Uil+1 ··· Uik ⊂ Um Therefore,

(Ui1 ··· Uil−1 )Uil (Uil+1 ··· Uik ) ⊂ UmUmUm ⊂ Um−1 ⊂ Un

Here we use the following inequalities il > m ≥ n + 1 mentioned above. This proves the lemma. Now we check the inclusions 1 1 (∗) B 1 (e) := {x ∈ G : ρ(e, x) < } ⊂ Un ⊂ B 1 [e] := {x ∈ G : ρ(e, x) ≤ } 2n+1 2n+1 2n 2n (I) If x ∈ B 1 (e) then by definition of ρ and Wr thee exists k ∈ N such that 2n+1 1 1 1 x ∈ Ui1 ··· Ui and + ··· + < . k 2i1 2ik 2n+1 By the lemma above this implies that x ∈ Un. 1 1 (II) Observe that Un ⊂ Wr for every r > 2n . This implies that ρ(x, e) ≤ 2n . 

Similarly can be obtained left invariant pseudometric with the property (*). How- ever, in general we cannot require that ρ simultaneously left and right invariant.

Theorem 8.2. (Pontryagin) Let G be a topological group such that G ∈ T0. Then G ∈ T 1 . 3 2

Proof. Since G ∈ T0 we already know that G ∈ T1. We have to prove that continuous real valued functions on G separate points and closed subsets. That is, for every g0 ∈ G and every closed subset g0 ∈/ F ⊂ G there exists a f : G → R such that f(g0) ∈/ cl(f(F )). Since G is a (semi)topological group it is equivalent to show this property only for g0 = e. Let F be a closed subset of G such that e∈ / F . Consider O := G \ F . By standard properties of topological groups there exists a sequence {Un}n∈N of symmetric nbds of e in G with U1 = O such that 3 Un+1 ⊂ Un for every n ∈ N. By Theorem 8.1 there is a right invariant pseudometric ρ on G such that B 1 (e) ⊂ Un ⊂ B 1 [e]. 2n+1 2n In particular, for n = 1 we have

B 1 (e) ⊂ O ⊂ B 1 [e]. 4 2 Now define the desired function f : G → R as f(x) := ρ(e, x). Then f separates e 1 and F because f(e) = 0 and f(x) ≥ 4 for every x ∈ F = G \ O. Finally note that f : G → R is continuous. Indeed, f is continuous at e by (*) of Theorem 8.1. It is enough for the continuity of f (at arbitrary g0 ∈ G) because ρ is right invariant.  Theorem 8.3. (Birkhoff-Kakutani Theorem) Let G be a Hausdorff topological group. The following are equivalent: (1) G is metrizable. (2) G ∈ B1 (first countable, that is has a countable basis at every point). 32

Proof. (1) ⇒ (2) is always true for topological spaces.

(2) ⇒ (1) There exists a countable basis γ := {Un}n∈N of open nbds at e. Again, by the standard properties of topological groups one may assume that every Un is 3 symmetric and Un+1 ⊂ Un for every n ∈ N. By Theorem 8.1 there exists a right invariant pseudometric ρ on G which satisfies (*). Since (G, τ) is Hausdorff then ∩nUn = {e}. By (*) we know that B 1 ⊂ Un. This implies that ∩nB 1 (e) = {e}. 2n+1 2n Hence, ρ(e, x) > 0 for every x 6= e. Therefore, ρ is a metric. The property (*) also implies that the topology of ρ is the given topology of G. Indeed, top(ρ) induces the same nbd system at e as the topology τ of G. At the same time, N(g0) = N(e)g0 for both of these topologies top(ρ) and τ (for the first, recall that ρ is right invariant). So, ρ is a metric on G which generates the original topology. This means that G is metrizable.  So, every Hausdorff topological group G is a Tychonoff space. Note that during many years it was an open question (of Pontryagin) if G is even normal. This question negatively was answered by Markov. His approach was much more important than above mentioned question. Markov introduced a concept of a free topological group F (X) over a given Tychonoff space X. He showed that X is closely embedded into F (X). This answers the question because the normality is a hereditary property with respect to closed subsets. It is enough to take G := F (X) for a Tychonoff space X which is not normal (for example, X= Sorgenfrey plane). Much later was proved that the Tychonoff topological group ZR is not normal.

8.1. Uniform spaces. Definition 8.4. (A. Weil) Let X be a set. A nonempty subset µ ⊂ P (X × X) (so, µ is a collection of relations on X) is said to be an uniform structure on X if the following conditions are satisfied: (1) ∆ ⊂ ∀ε ∈ µ. (2) ε−1 ∈ µ ∀ε ∈ µ. (3) ∀ ε ∈ µ ∃δ ∈ µ δ ◦ δ ⊂ ε. (4) ∀ ε1, ε2 ∈ µ ε1 ∩ ε2 ∈ µ. (5) δ ∈ µ and δ ⊂ ε ⇒ ε ∈ µ. (6) (”Hausdorff property”) ∩{ε : ε ∈ µ} = ∆. Then (X, µ) is said to be a uniform space. The concept of the uniform spaces is a far reaching simultaneous generalization of two important concepts: metric space and topological group. Let us say that a subset γ ⊂ µ is a uniform base of µ if for every ε ∈ µ there exists δ ∈ γ such that δ ⊂ ε. For every ε ∈ µ and every x0 ∈ X define an analogue of a ball in metric spaces as follows:

Bε(x0) = ε(x0) := {x ∈ X :(x0, x) ∈ ε} The induced topology top(µ) is defined as follows:

O ∈ top(µ) ⇔ (x0 ∈ O ⇒ ∃ε ∈ µ Bε(x0) ⊂ O) 33

Exercise 8.5. Show that (X, top(µ)) is a topological space for every uniform space (X, µ). Example 8.6. (1) For every metric space (X, d) the pair (X, µ(d)) is a uniform space, where 2 µ(d) := {R ⊂ X × X : ∃ε > 0 Rε ⊂ R},Rε := {(x, y) ∈ X : d(x, y) < ε}

(2) For every Hausdorff topological group G the pair (G, µr(G)) (the right uni- formity) is a uniform space, where 2 2 −1 µr(G) := {R ⊂ G : ∃U ∈ N(e) RU ⊂ R},RU := {(x, y) ∈ G : xy ∈ U}

(3) Similarly can be defined the left uniformity µl on G. 2 2 −1 µl(G) := {R ⊂ G : ∃U ∈ N(e) LU ⊂ R},LU := {(x, y) ∈ G : x y ∈ U} Remark 8.7. There are two more natural uniformities on every topological group G (which induce the given topology): two-sided uniformity µl∨r and the Roelcke uniformity µl∧r. Here µl∨r is defined as the sup (µl ∪ µr is a subbase) of left and right and µl∧r is the inf left and right. 34

9. Lectures 10,11,12 Let (X, µ) be a uniform space. We say that a subset γ ⊂ µ is a base of the uniform structure (or, a uniform base) of µ if for every ε ∈ µ there exists δ ∈ γ such that δ ⊂ ε. We say that γ is uniform prebase if γ∩fin is a base. 2 −1 For example, {RU }U∈N(e), RU := {(x, y) ∈ G : xy ∈ U} is a uniform base of the right uniformity µr on the topological group G. The union γ := {RU }U∈N(e) ∪ {LU }U∈N(e) is a prebase of the, so-called, two-sided uniformity on G. Example 9.1. Additional examples of uniform spaces: (1) On every set X the set

µ∆ := {ε ⊂ X × X : ∆ ⊂ ε} defines the maximal possible uniform structure on X which has the single point uniform base γ := {∆}. Clearly, top(µ∆) is discrete. (2) For every compact Hausdorff space (K, τ) the set µK of all nbds of ∆ in K ×K is a uniformity. Then top(µK ) = τ. It is the unique possible uniformity on K which is compatible with the topology. Note that for the natural metric uniformity µd on R not every nbd of ∆ is an element of µd. (3) For every system of pseudometrics F := {ρi}i∈I (which separates the points) on a set X we have the corresponding (Hausdorff) weak uniformity µF , where the uniform prebase is the collection

2 [i, ε] := {(x, y) ∈ X : ρi(x, y) < ε} where i ∈ I and ε > 0. (4) More generally. Let X be a set and fi : X → (Y, µi) is a set of functions into (pseudo)uniform spaces. Then the collection

2 [i, ε] := {(x, y) ∈ X :(fi(x), fi(y)) ∈ ε}

with i ∈ I and ε ∈ µi is a prebase of a (pseudo)uniformity on X which is called a weak uniformity. (5) Subspace and product of uniform spaces can be naturally defined (as a kind of weak uniformity). Note that the corresponding topologies are exactly subspace and product topologies. (6) Let (Y, µ) be uniform space, X a topological space. On F ⊂ Y X consider the uniformity µco (of compact convergence). Its uniform subbase is a system of subsets

[K, ε] := {(f1, f2) ∈ F × F :(f1(x), f2(x)) ∈ ε ∀x ∈ K} where ε ∈ µ and K is a compact subset of X. Then if F ⊂ C(X,Y ) we have top(µco) = τco. For details see for example [1]. Similarly can be defined the uniformity of the pointwise convergence. Here X for any F ⊂ Y we have top(µp) = τp. The collection [X, ε] defines the uniformity of uniform convergence on X. 35

The class of all uniform spaces define a category Unif of uniform spaces. Mor- phisms in Unif are the uniformly continuous functions. A function f :(X1, µ1) → (X2, µ2) is said to be uniformly continuous if

∀ε ∈ µ2∃δ ∈ µ1 :(f(x), f(y)) ∈ ε ∀(x, y) ∈ δ. On a compact space K every continuous map f : K → (Y, µ) into a uniform space is uniformly continuous with respect to the natural (uniquely defined) compatible uniformity on X. We have a forgetful functor Unif → T op. What is the range of this functor ? The following Proposition 9.3 gives the expected answer. Example 9.2. Let X be a Tychonoff space. for every continuous bounded function f ∈ Cb(X) define the induced pseudometric ρf (x, y) := |f(x) − f(y)|. The system {ρf : f ∈ Cb(X)} is a subbase of some uniformity µβ on X (see Examples 9.1 items (3) or (4)). Observe that since X is Tychonoff the corresponding topology top(µbeta) is the original topology of X. Proposition 9.3. A topological space (X, τ) admits a compatible uniform structure iff X is Tychonoff (i.e., X ∈ T 1 ). 3 2 first proof: for every continuous function f : X → R define the induced pseudometric ρf (x, y) := |f(x)−f(y)|. Observe that the corresponding weak uniformity on X induces the orig- inal topology of X.

second proof: X,→ K ∈ Comp2 iff X ∈ T 1 . 3 2 (Comp2 ⊂ T4 ⊂ T 1 . On the other hand, if X ∈ T 1 then there exists a collection 3 2 3 2 S ⊂ C(X) which separates points and closed subsets of X. We can suppose that S S ⊂ C(X, [0, 1]). Then the diagonal function fS : X → [0, 1] =: K ∈ Comp2 is a topological embedding as it follows by a classical lemma 9.4 of Tychonoff.

Lemma 9.4. Let S := {fi : X → Yi} be a set of some continuous functions. Consider Q the diagonal function fS : X → i∈I Yi. Then Q (1) If S separates points of X then fS : X → i∈I Yi is a continuous injection. (2) If S, in addition, separates points and closed subsets of X then fS : X → Q i∈I Yi is a topological embedding. Proof. (1) is trivial. (2) We have to show that the restricted map fS : X → fS(X) is open. It is enough to show that for every x0 ∈ X and every U ∈ N(x0) the image fS(U) contains an Q intersection of fS(X) with a neighborhood O of fS(x0) in i∈I Yi. By our assumption there exists fi0 : X → Yi0 such that fi0 (x0) ∈/ cl(fi0 (X \ U)). Now the desired O ∈ N(f (x )) can be defined as O := (π−1(cl(f (X \ U))))c. S 0 i0 i0  Q Remark 9.5. Note that for every topological product Y := i∈I Yi the set of all natural maps (generalized projections) Y Y πJ : Yi :→ Yj i∈I j∈J 36 with finite subsets J ⊂ I separates points and closed subsets of Y (as it follows from the definition of product topology). At the same time, observe that the set of all usual projections πi : Y → Yi i ∈ I does not separate the points and closed subsets in Y . For example, for R2 the projections do not separate the points and closed subsets. Indeed, take for example any circle and its center in R2.

We have also another natural functor Metr → Unif. Its range is the class of all metrizable uniform spaces. The following remark can be treated as a far reaching generalization of Pontryagin’s theorem. Remark 9.6. (Alexandrov-Urysohn) A uniform space (X, µ) is metrizable iff µ has a countable uniform base. Hint: One direction is trivial because for every metric space (X, d) the countable collection 2 1 R 1 := {(x, y) ∈ X : d(x, y) < } n n is a base of the corresponding uniformity generated by d. Second direction is similar to the proof of Pontryagin’s theorem.

µl(G) and µr(G) are the same for: abelian groups, compact groups. In general, they are different.

Example 9.7. Let G := GL2(R). Then µl(G) 6= µr(G). Proof. Sketch: Consider two sequences of matrices: n 1  1 1  x := and y := n n2 n 0 1 n 0 1  1 − 1  Then x−1 = n n and we have n 0 1 1 1 + 1  1 0 y x = y (x−1)−1 = n n2 → E = n n n n 0 1 0 1 1 1 + 1 1 1 x y = (x−1)−1y = n → A = n n n n 0 1 0 1 Choose disjoint nbds O ∈ N(e),W ∈ N(A) in G. Now observe that for every symmetric V ∈ N(e) there exists n = nV ∈ N such that

( −1 −1 (xn ) yn = xnyn ∈ W (9.1) −1 −1 −1 xn yn = (ynxn) ∈ V

( −1 L (x , yn) ∈/ Of (9.2) n −1 R (xn , yn) ∈ Vf

L This implies that Of ∈/ µr. Therefore, µl 6= µr.  37

For uniform spaces one may define a very successful analogue of Cauchy sequences, Cauchy filters, completeness ... For example, a sequence xn in a topological group is a Cauchy sequence for µr if −1 for every U ∈ N(e) there exists n0 ∈ N such that xnxm ∈ U for all n, m ≥ n0. More generally, a filter α on G is a Cauchy filter for µr if for every U ∈ N(e) there exists A ∈ α such that AA−1 ⊂ U. Completion of a Hausdorff topological group G with respect to the two-sided uni- formity admits a natural structure of a topological group (which contains G as a (dense) topological subgroup). Every compactification is a particular case of a completion. Namely, a completion of a uniform space (X, µ) is a compactification iff µ is totally bounded. Meaning that for every ε ∈ µ and ε-covering {ε(x)}x∈X there exists a finite subcovering. For every Tychonoff space X the uniformity µβ defined in Example 9.2 is totally bounded and its completion leads to the maximal (the so-called Chech-Stone) com- pactification X → β(X). Remark 9.8.

(1) For every uniform space (X, µ) denote by Unifb(X) the set of all bounded uniformly continuous functions X → R. Then Unifb(X) is a closed subalgebra of Cb(X) such that Unifb(X, µ) separates points and closed subsets. (2) For a topological group G and its right uniformity µr denote by RUC(G) the set Unifb(G, µr). Note that f ∈ RUC(G) iff ∀ ε > 0 ∃U ∈ N(e) |f(ux) − f(x)| < ε ∀u ∈ U, x ∈ G 9.1. Theorems of Teleman. When a topological group G can be represented on a Banach space by linear isometries. That is, when G,→ Iso(V ) ? In fact Teleman’s theorem below 9.10 shows that always.

Every finite group G can be represented on Rn with n = |G|. Indeed,take n G,→ Sn ,→ Isolin(R ) = On(R). Exercise 9.9. Every discrete group G admits an effective isometric representation on lG and also on the Hilbert space l2(G).

For example, on l∞ := l(N) and on l2 := l2(N) if G is countable. Note that for n |G| = n the Banach space lG is a copy of (R , || · ||max). What happens for nondiscrete topological groups ? Our aim is to prove the follow- ing theorem of Teleman (rediscovered by many authors). Theorem 9.10. (Teleman’s theorems) Let G be a Hausdorff topological group. Then

(1) G is embedded into Isolin(V ) for some Banach space V . (2) G is embedded into Iso(M, d) for some metric space (M, d). (3) G is embedded into Homeo(K) for some compact space K. Clearly, (1) ⇒ (2). Teleman’s theorem suggests more challenging question “how good” can be the target Banach space V ? This questions leads to a natural hierarchy of topological groups closely related to a hierarchy of main-stream Banach space classes like: Hilbert, reflexive, Asplund, Rosenthal, etc. 38

Remark 9.11. For example, the topological group Homeo+([0, 1]) (of all orienta- tion preserving homeomorphisms) is not reflexively, or, even, Asplund, representable. However it is Rosenthal representable. It is an open question if every separable metrizable topological group (enough to examine G := Homeo([0, 1]N)) is Rosenthal representable. First we provide here some related definitions and results. With every Banach space V one may naturally associate several structures which are related to: Analysis, topological dynamics, topological (semi)groups. Let V be a Banach space. By V ∗ we denote its dual space. That is, the vector space of all continuous linear functionals f : V → R. For every f ∈ V ∗ one may define its operator norm ||f|| := sup{|f(v)| : ||v|| ≤ 1}. Then (V ∗, || · ||) becomes a Banach space and the canonical bilinear function ∗ V × V → R, (v, f) 7→< v, f >:= f(v) satisfies the following | < v, f > | ≤ ||v|| · ||f|| for every (v, f) ∈ V × V ∗. In particular, that function is continuous. By L(V,V ) (or, L(V )) we denote the set of all continuous linear maps V → V .

Definition 9.12. The weak topology τw on V is the weak topology on the set V with respect to the set V ∗ of all continuous linear functionals f : V → R. A net vi converges to v in this topology iff the net f(vi) in R converges to f(v) for all f ∈ V ∗. The collection ∗ {[v0; f1, ··· , fn; ε]: ε > 0, fi ∈ V ∀ i ∈ {1, ··· , n}} is a local topological base at v0, where, as usual

[v0; f1, ··· , fn; ε] := {v ∈ V : |fi(v0) − fi(v)| < ε ∀ i ∈ {1, ··· , n}}. Weak topology is weaker than the norm topology on V and these two topologies coincide iff V is finite dimensional. Definition 9.13. The weak-star topology on V ∗ is the weak topology on the set V ∗ with respect to the set V ⊂ V ∗∗ of continuous linear functionals ∗ v : V → R, f 7→ f(v), v ∈ V. A net fi converges to f in this topology iff the net fi(v) in R converges to f(v) for all v ∈ V . The collection

{[f0; v1, ··· , vn; ε]: ε > 0, vi ∈ V, ∀ i ∈ {1, ··· , n}} is a local topological base at f0, where, ∗ [f0; v1, ··· , vn; ε] := {f ∈ V : |f0(vi) − f(vi)| < ε ∀ i ∈ {1, ··· , n}}. 39

Weak-star topology is weaker than the weak topology on V ∗ and these two topolo- gies coincide iff V is finite dimensional. One of the important properties of the weak-star topology is Alaouglu theorem which asserts that every bounded weak-star closed subset of the dual V ∗ is compact for every Banach space V . In particular, it is true for the closed unit ball B∗ := {f ∈ V ∗ : ||f|| ≤ 1}. Definition 9.14. Let V be a Banach space.

(1) The strong operator topology τsot on L(V,V ) is the pointwise topology inherited V from (V, || · ||) . That is, a net si converges to s iff si(v) converges to s(v) in the norm topology for every v ∈ V . The collection

{[s0; v1, ··· , vn; ε]: ε > 0, vi ∈ V, ∀ i ∈ {1, ··· , n}}

is a local topological base at s0 ∈ L(V,V ), where, as usual

[s0; v1, ··· , vn; ε] := {s ∈ L(V,V ): ||svi − s0vi|| < ε} ∀ i ∈ {1, ··· , n}} (2) Replacing the norm topology of V by its weak topology we obtain the weak operator topology τwot on L(V,V ). A net si τw-converges to s in L(V,V ) iff f(si(v)) converges to f(s(v)) in R for every v ∈ V . The collection ∗ {[s0; v1, ··· , vn; f1, ··· , fn; ε]: ε > 0, vi ∈ V, fi ∈ V ∀ i ∈ {1, ··· , n}}

is a local topological base at s0 ∈ L(V,V ), where, as usual

[s0; v1, ··· , vn; f1, ··· , fn; ε] := {s ∈ L(V,V ): ||fi(svi)−fi(s0vi)|| < ε} ∀ i ∈ {1, ··· , n}}

We use the notation: Θ(V )s, Iso(V )s (respectively, Θ(V )w, Iso(V )w) or simply Θ(V ) and Iso(V ), where the topology is understood. Proposition 9.15. (1) For every compact space Y the semigroup C(Y,Y ) endowed with the compact open topology τco is a topological monoid. (2) Note also that the subset Homeo (Y ) in C(Y,Y ) of all homeomorphisms Y → Y is a topological group. (3) For every submonoid (S, τco) ⊂ C(Y,Y ) the natural monoid action (S, τco) × Y → Y is continuous. (4) Furthermore, it satisfies the following remarkable minimality property. If τ is an arbitrary topology on S such that (S, τ) × Y → Y is continuous then τco ⊂ τ. Proof. (1) and (2) are parts of a homework. (3) The continuity of the action (S, τco) × Y → Y is easy taking into account that the compact open topology on S ⊂ C(Y,Y ) is the same as the topology of compact convergence. So, s0 and s in S are ε-close means that (s0y, sy) is ε-close for every y ∈ Y (where ε is an element of the uniformity µY on Y ). Since s0 ∈ C(Y,Y ), we can choose δ ∈ µY such that (s0y0, s0y) ∈ ε for every (y0, y) ∈ δ. Therefore, we can “control” (s0y0, sy) in the entourage ε ◦ ε (“triangle equality” axiom of uniform spaces). 40

(4) Let (S, τ) × Y → Y be continuous. Then by the compactness of Y it is easy to see the following

∀s0 ∈ S ∀ε ∈ µY ∃U ∈ Nτ (s0):(s0y, sy) ∈ ε ∀y ∈ Y.

This proves that the topology of compactness convergence τco is weaker than τ.  Proposition 9.16. (1) For every metric space (M, d) the semigroup Θ(M, d) of all d-contractive maps f : X → X (that is, d(f(x), f(y)) ≤ d(x, y)) is a Hausdorff topological monoid with respect to the topology of pointwise convergence. The group Iso(M) of all onto isometries is a topological group. (2) Furthermore, the evaluation map S × M → M is a jointly continuous action for every subsemigroup S ⊂ Θ(M, d). (3) For every normed space (V, || · ||) the semigroup Θ(V ) of all contractive linear operators V → V endowed with the strong operator topology (being a topolog- ical submonoid of Θ(V, d) where d(x, y) := ||x − y||) is a topological monoid. The subspace Iso(V ) of all linear onto isometries is a topological group. Proof. (1) is a part of a homework. (2) The continuity of the action S ×M → M at the point (s0, v0) is straightforward using the following inequality

d(s0v0, sv) ≤ d(s0v0, sv0) + d(sv0, sv) ≤ d(s0v0, sv0) + d(v0, v) (3) easily follows from (1).  An action S × X → X on a metric space (X, d) is contractive if every s-translation s˜ : X → X lies in Θ(X, d). It defines a natural homomorphism h : S → Θ(X, d). Remark 9.17. If an action of S on (X, d) is contractive then it is easy to show that the following conditions are equivalent: (i) The action is continuous. (ii) The action is separately continuous. (iv) The natural homomorphism h : S → Θ(X, d) of monoids is continuous.

Proposition 9.18. Θ(V ) is a semitopological monoid with respect to the weak oper- ator topology.

Proof. An exercise. 

Remark 9.19. The semitopological semigroup Θ(V )w is compact iff V is a reflexive Banach space. Hint: Observe that Θ is closed in (B, w)B for every V . Then use the following characterization of reflexivity: V is reflexive iff every bounded weakly closed subset is weakly compact iff B := BV is weakly compact. For every semigroup (S, ·) one may define its opposite semigroup Sop with the “opposite multiplication” (x, y) 7→ y · x. Clearly, the usual inverse map on any group G defines an isomorphism between G and Gop. For any left action S × X → X we have the induced right action X × S → X and vice versa. 41

Theorem 9.20. For every normed space V and a strongly continuous homomorphism h : S ⊂ Θ(V )op the induced action S × B∗ → B∗ on the compact space B∗ is jointly continuous.

∗ ∗ Proof. For the continuity of the action S × B → B at the point (s0, f0) use the following inequaity

|(s0f0)(vi) − (sf)(vi)| ≤ |(s0f0)(vi) − (s0f)(vi)| + |(s0f)(vi) − (sf)(vi)| ≤

|f0(vis0) − f(vis0)| + ||f|| · ||vis0 − vis|| 

Every left action π : S × X → X induces the right action C(X) × S → C(X) (where (fs)(x) = f(sx)) and a co-homomorphism hπ : S → Θ(C(X)) and. While the translations are continuous, the orbit maps f˜ : S → C(X), s 7→ fs are not necessarily norm (even weakly) continuous and require additional assumptions for their continuity. It turns out that this happens iff f mimics the property of right uniform continuity like RUC(G) (see Remark 9.8). For every normed space V the usual adjoint map adj : L(V ) → L(V ∗), σ 7→ σ∗, < vs, u >=< v, s∗u > is an injective co-homomorphism of monoids. For simplicity we write s instead of s∗. Theorem 9.21. (Teleman’s fisrt theorem) Every Hausdorff topological group G can be embedded into Isolin(V ) for some Banach space V . Proof. Consider the Banach space V := RUC(G) of all right uniformly continuous bounded functions which is a closed linear subspace of (Cb(G), ||·||sup). As we already mentioned in Remark 9.8 the set RUC(G) separates points and closed subsets of G. The usual left action G × G → G induces the following right action V × G → V, (f, g) 7→ fg on V := RUC(G). Then this action is linear contractive (by linear isometries) and continuous. The linearity is trivial. For the contractivity observe that clearly, ||fg|| ≤ ||f|| for all g ∈ G. Since G is a group in fact we get ||fg|| ≤ ||f||. It is easy to show the continuity of that linear action using Remark 9.8.2. Indeed the condition f ∈ RUC(G) means that

∀ ε > 0 ∃U ∈ N(e) |f0(ux) − f0(x)| < ε ∀u ∈ U, x ∈ G it implies that

||f0u − f0|| ≤ ε ∀u ∈ U

Now in order to see the continuity of that action at point (f0, g0) use the following inequality

||f0g0 − fg|| = ||f0g0 − fg0u|| ≤ ||f0g0 − f0g0u|| + ||f0g0u − fg0u|| =

||f0g0 − f0g0u|| + ||(f0 − f)g0u|| = ||f0g0 − f0g0u|| + ||f0 − f||. 42

Consider the natural homomorphism op h : G → Isolin(V ) , h(g)(v) := vg. This map is well defined because the action is contractive. Also h is a homomor- op phism. Clearly, (Isolin(V ) , τsot) is a Hausdorff topological group. So, it is enough to show that h is a topological embedding. Continuity of h. By the trivial “transport argument” it is enough to show the continuity of h at e ∈ G. Consider the following typical basic nbd of h(e) = id in the op topological group (Isolin(V ) , τsot): op [id; f1, f2, ··· , fn; ε] := {s ∈ Isolin(V ) : ||fi − fis|| < ε ∀i ∈ {1, ··· , n}}

Note that here fi ∈ V play the role of vi in the definition of strong operator topology. Since each fi is in RUC(G) we can find U ∈ N(e) such that ||fiu − fi|| < ε for every u ∈ U and every i ∈ {1, ··· , n}. Then clearly, g ∈ U implies that h(g) ∈ [id; f1, f2, ··· , fn; ε]. This proves the continuity of h (at e).

This map h is injective because if g1 6= g2 then f(g1) 6= f(g2) for some f ∈ RUC(G). Therefore, h(g)(f1) = f1g 6= f2g = h(g)(f2). Now we show that, in fact, h is a topological embedding. It is enough to show that the homomorphism h−1 : G0 → G is continuous (at id), where G0 := h(G). For a given U ∈ N(e) there exists f ∈ RUC(G) such that f(e) ∈/ cl(f(G \ U)) in R. Then there exists ε > 0 (take ε := d(e, G \ U)) such that |f(g) − f(e)| < ε ⇒ g ∈ U. Now we get ||h(g)(f) − f|| = ||fg − f|| < ε ⇒ g ∈ U. Or, equivalently, h(g) ∈ [id; f; ε] ⇒ g ∈ U. −1 0 This means that the function h : G → G is continuous at id.  9.2. Appendix. Lemma 9.22. For every normed space V the injective map op ∗ ∗ γ : Θ(V )s ,→ C(B ,B ) induced by the adjoint map adj : L(V ) → L(V ∗), is a topological (even uniform) monoid embedding. In particular (by Proposition 9.15.3), Θ(V )op × B∗ → B∗ op ∗ is a jointly continuous monoidal action of Θ(V )s on the compact space B . Proof. The strong uniformity on Θ(V ) is generated by the family of pseudometrics {pv : v ∈ V },where pv(s, t) = ||sv − tv||. On the other hand the family of pseudo- ∗ metrics {qv : v ∈ V },where qv(s, t) = sup{|(fs)(v) − (ft)(v)| : f ∈ B } generates ∗ ∗ the natural uniformity inherited from C(B ,B ). Now observe that pv(s, t) = qv(s, t) by the Hahn-Banach theorem. This proves that γ is a uniform (and hence, also, topological) embedding.  43

For every compact space K we have a topological embedding (Gelfand representa- tion) ∗ δ : K,→ C(K) , x 7→ δx where δx : C(K) → R is the standard “evaluation at x functional” defined as δ(f) := f(x). It can be identified with the “point measure” on K. Theorem 9.23. Let S × K → K be a continuous action. Then (1) the induced action S×B∗ → B∗ is continuous on B∗ ⊂ V ∗, where V := C(K). (2) Moreover, if S carries the compact open topology inherited from C(K,K) then ∗ ∗ op the homomorphisms S → C(B ,B ) and h : S → Θ(V )s are topological embeddings. Proof. (1) The induced right linear action C(K) × S → C(K) is continuous (because the orbit maps are norm continuous). This action is contractive. It follows that h : S → Θ(V )op is a well defined strongly continuous homomorphism. Moreover, B∗ is an S-subset under the dual action. By Theorem 9.20 we obtain that the action S × B∗ → B∗ is continuous. (2) Let S ⊂ C(K,K). Denote by τ0 the induced topology on S. The action S × K → K can be treated as a restriction of the bigger action S × B∗ → B∗, where K naturally is embedded into B∗ via Gelfand’s map. Then the topology τ on S ∗ ∗ inherited from C(B ,B ) majors the original topology τ0. Hence, τ0 ⊂ τ. On the other hand, the continuity of S × B∗ → B∗ easily implies by Proposition 9.15 that τ ⊂ τ0 on S. Summing up we conclude that τ = τ0 on S.  Theorem 9.24. (Teleman’s second theorem) Every Hausdorff topological group G is embedded into Homeo(K) for some compact space K. Proof. Combine Teleman’s first theorem 9.21 and Lemma 9.22. It follows that the ∗ ∗ desired compact space K we can choose as (B , τw∗ ), where B is the closed unit ball of the dual space V ∗ of V := RUC(G), where B∗ is endowed with the weak- ∗ star topology. Recall that by Alaouglu theorem, (B , τw∗ ) is always compact for any Banach space V . 

References 1. J. Kelley, General Topology. 2. G. Lukacs, Compact-like topological groups, 2009. 3. J. Munkres, Topology. 4. S. Morris, Topology without tears, 2011. 5. S. Morris, Pontryagin duality and the structure of locally compact abelian groups, 1977. 6. V. Pestov, Topological groups: where to from here, 2000. 7. A. Wilansky, Topology for Analysis, 1998.

Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel E-mail address: [email protected] URL: http://www.math.biu.ac.il/∼megereli