Function Algebras Over Groups

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Function Algebras Over GroupsBy Dr.V.MuruganandamPondicherry University


<p>FUNCTION ALGEBRAS OVER GROUPSV. MURUGANANDAM</p> <p>Abstract. One of the primary aspects of harmonic analysis is</p> <p>the study of functions on a homogenous space by means of suitablegroup actions. The other end of this thread is to introduce various</p> <p>classes of functions on groups and study their properties vis-a-vis the underlying groups. In this compact course, we focus on this We concept with particular reference to the most useful and widely studied notion in harmonic analysis namely Amenability. aim to look beyond the amenability at the end of the course.</p> <p>1.</p> <p>Preliminaries</p> <p>Denition 1.1. A topological space G is said to be a topological group if it is a group and the map (x, y) xy1 is continuous from G G into G.Let us give some important classes of locally compact Hausdor groups.</p> <p>Examples 1.2. (i) Rn is a locally compact group. (ii) For every n 1, Tn is a compact group. (iii) For every xed n, let GL(n, R) be the group consisting of all n n invertible matrices with real entries. Then GL(n, R) is a locally compact group as it is an open subset of Rn2 . (iv) SL(n, R) = {x GL(n, R) : det(x) = 1} and O(n) consisting of orthogonal matrices are also locally compact groups as they form closed subsets of GL(n, R). a b (v) S = x = 0 1 : a &gt; 0, b R . (vi) If G is the Heisenberg group given by 1 x z 0 1 y : x, y, z R g= 0 0 10This notes is based on a series of lectures given in the Workshop and a centenaryconference on Analysis and Applications, IISc Mathematics Initiative (IMI), I.I.Sc, Bangalore, from May, 14-23, 2009. writing down the notes.1</p> <p>The author thanks R. Lakshmi Lavanya for</p> <p>2</p> <p>V. MURUGANANDAM</p> <p>then G is noncompact and nonabelian. (vii) Let F2 denote the free group generated by a, b, with ab = ba. It is a nonabelian discrete group. Arbitrary element of this group is of the form am bn ak where m, n, k belong to Z.Let tively. Let</p> <p>Cc (G)</p> <p>and</p> <p>Cb (G)</p> <p>denote the space of all continuous functions</p> <p>with compact support and continuous bounded functions on</p> <p>G, respec-</p> <p>Clu (G), Cru (G) and Cu (G) denote the space of all left uniformly G.</p> <p>continuous functions, right uniformly continuous functions and uniformly continuous functions on 1.1.</p> <p>Measure algebra. Denition 1.3. Let X be a locally compact space. A Borel measure is said to be regular if (1) (K) &lt; , for every compact set K. (2) For every Borel set E , (E) = inf{ (U ) : U is open and E (3)</p> <p>U }. If E is any Borel set and (E) &lt; , then (E) = sup (K) : K is compact, K E .</p> <p>Denition 1.4. A Banach space A over C is called a Banach algebra if A is an algebra satisfyingxy x y , x, y A</p> <p>and is called Banach -algebra if it admits an involution x x on A such that x = x for all x in A.Let</p> <p>G</p> <p>be a locally compact Hausdor group. If</p> <p>M (G)</p> <p>denotes the</p> <p>space of complex Borel measures on</p> <p>G,</p> <p>then it forms a Banach Space.</p> <p>In what follows, we briey show that the underlying group structure gives rise to two additional structures on</p> <p>M (G),are in</p> <p>with respect to which</p> <p>M (G)</p> <p>forms an Banach</p> <p>algebra. If</p> <p>, </p> <p>M (G),</p> <p>then dene</p> <p>(x) d( )(x) = is called the</p> <p>(xy) d(x) d(y).For every</p> <p>belongs to</p> <p>Recall that if</p> <p>x.</p> <p>If</p> <p>M (G) and . x belongs to G, then x denotes x, y are in G, then d(x y ) =</p> <p>convolution of the measures.</p> <p> and , </p> <p>the Dirac measure at</p> <p>(uv) dx (u) dy (v) = (xy) =</p> <p> dxy .</p> <p>3</p> <p>Therefore,</p> <p>M (G).</p> <p>The involution</p> <p>x y = xy . See </p> <p>that on</p> <p>e is the multiplicative M (G) is dened by (x1 ) d(x). M (G).</p> <p>identity of</p> <p> d =It is easy to see that 1.2.</p> <p>e</p> <p>is the identity for</p> <p>Group algebra.</p> <p>One of the milestones in the history of abstract</p> <p>Harmonic analysis was the existence of left invariant measure on a general locally compact Hausdor group that reads as follows. A Borel measure Borel</p> <p>m on G is said to be left invariant if m(xE) = m(E) for every set E, and for all x in G.</p> <p>Theorem 1.5. (Haar, Von Neumann) Let G be a locally compact Hausdor group. There exists a non-zero, positive, left invariant, regular Borel measure on G. Moreover, it is unique up to a positive constant.Such a measure is called Haar measure and the corresponding integral is called Haar integral. Loomis[11] for a proof. Let We refer to the books by A. Weil[18], always denote the Haar integral.</p> <p>dx</p> <p>Let us see some examples of Haar measures. (1) For any subset</p> <p>E</p> <p>of a discrete group, if</p> <p>m(E)</p> <p>is dened to be</p> <p>the number of elements in</p> <p>counting measure.</p> <p>E,</p> <p>then it is a Haar measure called</p> <p>(2) The Lebesgue measure is the Haar measure for the group</p> <p>Rn ,</p> <p>as it is translation invaraint. 2 1 (3) For the Torus T, f (ex )dx is the Haar integral. 2 0 1 (4) If G = GL(n, R) then verify that dg11 dg12 dgnn denes (det g)n a Haar measure. 1 (5) If G denotes the group given in 1.2(v) then 2 dadb is the Haar a measure. (6) The Lebesgue measure</p> <p>dg = dxdydzis the Haar measure for the Heisenberg group given in 1.2(v). Let</p> <p>Lp (G), 1 p &lt; </p> <p>of all measurable functions one denes L (G).</p> <p>denote as usual, the Banach space consisting f such that G |f (x)|p dx is nite. Similarly</p> <p>Theorem 1.6. (Lusin). For every p, 1 p &lt; , Cc (G) is dense inLp (G).</p> <p>Proposition 1.7. If f in Lp (G), 1 p &lt; is xed then the map x x f from G into Lp (G) is continuous.</p> <p>4</p> <p>V. MURUGANANDAM</p> <p>f Cc (G). Then, f is uniformly continuous. f. Let &gt; 0 be given. Since G is locally compact, there exists a neighborhood W of identity e such that W is compact. Choose a neighborhood U of e such that U W and if x, y G are such that x U y thenLet us assume that Let K be the support of</p> <p>Proof.</p> <p>|f (x) f (y)| &lt; being the xed Haar measure. Then for every</p> <p>(KW )</p> <p>,</p> <p>x, y G</p> <p>such that</p> <p>x Uy</p> <p>, we have</p> <p>||x f y f ||p 0 Since x (x) is continuous at e there x V0 (x) &lt; . For every &gt; 0 , (f ) =G</p> <p>V0</p> <p>f (x)(x) dx |f (x)| (x) dxV</p> <p>&lt; =Hence</p> <p>.V</p> <p>|f (x)| dx.</p> <p>. |L1 (G) is non-</p> <p> (f ) I </p> <p>in strong operator topology and so</p> <p>degenerate.</p> <p>1 Conversely suppose that : L (G) BL(H) is a non-degenerate 1 -representation of L (G). 1 Set K = [(L (G)(H)]. Then K is dense in H. Dene</p> <p>(x)((f )) = (x f )().Since</p> <p>x f f x f,</p> <p>for all</p> <p>f L1 (G),</p> <p>we have</p> <p>(x)((f )) = lim (x f )</p> <p>(f )()</p> <p>Therefore by 1.18, as an operator on</p> <p>(f )() .</p> <p>(x) is bounded on K. Hence (x) gets extended H. It is easy to see that (xy) = (x) (y). Since</p> <p>10</p> <p>V. MURUGANANDAM</p> <p> = (x1 )(x)() (x) , f, g in L1 (G), (f ) (g) = (f g)() =G</p> <p>is unitary. Since for every</p> <p>f (y)(y g) f (y)(y)((g))dyG</p> <p>=</p> <p>= (f )((g)) we have</p> <p>(f ) = (f ). </p> <p> Remark 1.20. For example , corresponding to is given by left convolution operators, since (f )(g)(x) =G</p> <p>(y)(g)(x)f (y)dy =G</p> <p>g(y 1 x)f (y)dy = f g(x).</p> <p> This representation is faithful in the sense that if (f ) = 0 then f = 0 1 in L (G).2.</p> <p>Positive definite functionsn</p> <p>Denition 2.1. A function : G C is said to be positive denite if for all c1 , c2 , ..., cn C and x1 , x2 , ..., xn G,ci cj (x1 xi ) 0. ji,j=1</p> <p>Example 2.2. If (, H) is any unitary representation of G then for any H function = , is positive denite since for any choice of xi and ci as in the denitionn</p> <p>ci cj (x1 xi ) = ji,j=1</p> <p>, </p> <p>where =</p> <p>i ci (xi )().</p> <p>Note that, taking n=1 and</p> <p>c1 = 1 in Denition 2.1, we have (e) 0.</p> <p>Proposition 2.3. Let be a positive denite function. Then(1) (2)</p> <p>(x1 ) = (x) |(x)| (e), x G.By hypothesis, the matrix</p> <p>Proof.</p> <p>nite. Consider</p> <p>A=</p> <p>((x1 xj ))1i,jn is positive semi dei (e) (x) . Since A = A , and det(A) 0, (x1 ) (e)</p> <p>the result follows.</p> <p>11</p> <p>Theorem 2.4. Let be a continuous function on G. Then the following are equivalent (1) is positive denite. (2) is bounded and , f f 0, f Cc (G).(3)</p> <p>, </p> <p> 0, M (G).</p> <p>Proof.</p> <p>nIf is a measure with nite support, that is if</p> <p> =i=1</p> <p> i x i ,</p> <p>observe that</p> <p>n</p> <p>, =i,j=1Assume that</p> <p>ci cj (x1 xi ). j</p> <p>, 0. Let f belong to Cc (G). Then there exists { } of measures with nite support such that converges to f (y)dy in the weak -topology. is positive denite. Then Therefore, (1) implies (2). Let us prove (2) implies (3). support. Then for any If to Let us suppose that</p> <p>has compact</p> <p>f in Cc (G) the function f belongs to Cc (G). {f } is a bounded approximate identity such that each f belongs Cc (G), then f belongs to Cc (G) and it converges to in the-topology. Therefore (3) is true in this case. belongs to</p> <p>weak If</p> <p>pact support such that</p> <p>M (G), then there exists { } in M (G) with com{ } converges to in the weak - topology.</p> <p>Therefore (2) implies (3). It is trivial that (3) implies (1).</p> <p>Notation:P (G).</p> <p>Let</p> <p>functions on</p> <p>G.</p> <p>P (G) denote If belongs</p> <p>the set of all continuous positive denite to</p> <p>P (G)</p> <p>then observe that</p> <p>belongs to</p> <p>Remark 2.5. In Example 2.2 we have seen that any matrix coecient belonging to a unitary representation is positive denite. Now we shall show that these are the only positive denite functions. In other words, we show that if belongs to P (G), then there exists a cyclic representation (, H) with cyclic vector such that (x) = , .2.1.</p> <p>GNS construction.</p> <p>Theorem 2.6. Let be any continuous positive denite function on G. Then there exists a cyclic representation (, H) with the cyclic vector such that (x) = (x)u, u locally almost everywhere.</p> <p>12</p> <p>V. MURUGANANDAM</p> <p>Proof.</p> <p>Dene</p> <p>, </p> <p> on</p> <p>L1 (G)</p> <p>by</p> <p>f, g</p> <p>= g f</p> <p>=G</p> <p> (x1 y)g(x)f (y)dxdy.</p> <p>1 It is easy to see that it denes a sesquilinear form on L (G). If N = 1 {f L (G) : f, f = 0} , one can see by Cauchy Schwarz inequality 1 one can see that N = {f L (G) : f, g = 0g L1 (G)} . There1 fore it forms a closed subspace of L (G).Moreover, since</p> <p>x f,xwe see that</p> <p>g</p> <p>= f, g</p> <p>(2.1)</p> <p>is invariant under left translation. 1 Let H0 denote the quotient space L (G)/N . Complete it to get a to</p> <p>N</p> <p>Hilbert space</p> <p> f belongs</p> <p>H. We shall dene L1 (G)/N take</p> <p>a representation</p> <p>on</p> <p>H</p> <p>as follows. If</p> <p> (x)(f ) = x1 f =Then</p> <p>x1 f</p> <p> G.</p> <p>extends to a unitary representation of</p> <p>Let us show that (, H) is cyclic. If {f } is a bounded approximate 1 identity of L (G) then take a subnet if necessary to conclude that f converges to a vector</p> <p>weakly in</p> <p>H.</p> <p>Then</p> <p> f, Since</p> <p> = lim f , f </p> <p>= lim f f, = G</p> <p>f (x)(x)dx.and</p> <p>(2.2)</p> <p>g, f </p> <p>=</p> <p>G</p> <p>G</p> <p> (x1 y)f (x)g(y)dxdy</p> <p>(x1 y)g(y)dy =Gwe see by (2.2) that</p> <p>(y)g(xy)dy = (x1 ), gG</p> <p>= g , (x) </p> <p>g, f </p> <p>=G</p> <p> f (x) g , (x) g , (f )() H.(2.3)</p> <p>=Hence</p> <p>[(L1 (G))]</p> <p>is total in</p> <p>and so the representation is cyclic.</p> <p>Finally,</p> <p> , f = lim f f , = lim f , (f ) Therefore, by (2.2),</p> <p>= , (f )()</p> <p>G</p> <p>f (x)(x)dx = (f )(), </p> <p>Remark 2.7. Using the preceding theorem, we conclude that the vector space B(G) dened in 1.16 is in fact linear span of continuous positive denite functions.</p> <p>13</p> <p>3.</p> <p>C</p> <p>algebras of groupsx A. C(X)of</p> <p>Denition 3.1. An Banach - algebra A is said to be C -algebra if the involution of A satises the additional conditionx x = x 2,</p> <p>Remark 1.</p> <p>Let</p> <p>X</p> <p>be a compact Hausdor space. The space</p> <p>X is a unital Banach algebra f f is an involution that makes C(X) into a C -algebra. Similarly if X is a locally compact noncompact Hausdor space then C0 (X) consists of continuous functions which vanish at innity forms a C -algebra without identity. 2. L (X, d) for any measure , is a C -algebra. 3. Let H be a Hilbert space. Then the unital Banach algebra BL(H) is a C algebra with the operator norm and the involution given by the map T T . In general any norm closed -subalgebra of BL(H) is a C -algebra.continuous complex valued functions on with the uniform norm. The map</p> <p>Theorem 3.2. If is dened on L1 (G) by f = sup { (f ) : is any non-degenerate -representation} (3.1) then it denes a norm on L1 (G). Moreover, the completion of L1 (G) with respect to this norm is a C -algebra. Proof. Clearly,f + gSuppose that</p> <p> || f</p> <p>+ g</p> <p>= 0. Then (f ) = 0 where : L1 (G) BL(L2 (G)) is the left regular representation. As is a faithful representation, we have f = 0. Therefore . is indeed a norm. f</p> <p>Denition 3.3. The C -algebra obtained above is called full C of G and is denoted by C (G).The following theorem gives yet another way to realize us rst recall</p> <p>-algebra</p> <p>C (G).</p> <p>Let</p> <p>Theorem 3.4. A C -algebra has suciently many irreducible representations to separate points of A. That is, for every x A, x = 0 there exists an irreducible representation of A such that (x) = 0. Theorem 3.5. let A denote the C (G)-algebra obtained by completing L1 (G) with where f = sup (f ) : G . Then C (G) and A are isometrically isomorphic</p> <p>14</p> <p>V. MURUGANANDAM</p> <p>Proof.</p> <p> : (L1 (G), . ) (A, . ) . Then (f ) f . So extended into a -homomorphism from C (G) into A. We claim that is injective. Suppose (f ) = 0, f C (G). Then (f ) = 0, G. By the preceding Gelfand-Raikov Theorem, f = 0. Therefore is injective. Since an injective -homomorphism between C -algebras isDene an isometry, we have</p> <p>(f ) = f ,Therefore,</p> <p> f C (G).Thus</p> <p>(C (G)) is closed in A. C -completion of (L1 (G), . )</p> <p>(C (G)) = A.</p> <p>Hence</p> <p>A</p> <p>is</p> <p>Proposition 3.6. Suppose that the group G is abelian. Then the full C -algebra of G is identied with C0 (G).Proof.If</p> <p>(, H)</p> <p>is a unitary irreducible representation of</p> <p>G</p> <p>then by</p> <p>Schur's lemma it is given by one-dimensional representation. That is, there exists a character going to Now</p> <p>identies</p> <p> such that (x) = (x), for every x G. G with the dual group given in Section 1. f (x) (x) dx. = f ( ), G</p> <p>(f ) =Gwhere</p> <p>f (x)(x) dx. =Therefore,</p> <p> = (x1 ). f G</p> <p>= sup (f ) = sup f ( ) = f G</p> <p> , since</p> <p> G. </p> <p>C (G) is the completion of (L1 (G), . ) and (L1 (G)) = G, we have C (G) is the completion of {f C0 (G) : . norm}. Since {f : f Cc (G)} is dense in L1 (G), we have C (G) = C0 (G).Since</p> <p>Remarks 3.7. 1. If is any -representation of L1 (G), then gets extension to a -representation of C (G). 2. If 1 and 2 are two non-degenerate representations of L1 (G) then they are equivalent if and only if their extensions to C (G) are equivalent. 3. Summarizing by we observe by Theorem 1.19 and the preceding theorem that there is a bijective correspondence between the unitary representations of G and non-degenerate -representations of C (G) such that irreducible ones go into irreducible ones. Moreover this identication respects equivalence relation among the representations.</p> <p>15</p> <p>From the proof of Theorem 3.2 we observe that instead of taking all unitary representations in the equation (3.1), if we take the left regular representation alone, we get another</p> <p>C</p> <p>-algebra.</p> <p>Denition 3.8. The closure of -subalgebra (L1 (G)) in BL(L2 (G)) is a C -algebra and is called reduced C -algebra of G. We denote this C -algebra by...</p>


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