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1.
We prove that the crossed product C*-algebra C*r(, ) of a freegroup with its boundary sits naturally between the reducedgroup C*-algebra C*r and its injective envelope I(C*r). In otherwords, we have natural inclusion C*r C*r(, ) I(C*r) of C*-algebras.  相似文献   

2.
Let = 2cos (/5) and let []. Denote the normaliser ofG0() of the Hecke group G5 in PSL2() by N(G0()). Then N(G0())= G0(/h), where h is the largest divisor of 4 such that h2 divides. Further, N(G0())/G0() is either 1 (if h = 1), 2 x 2 (if h= 2) or 4 x 4 (if h = 4).  相似文献   

3.
Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat vm–1 ... = 1, where v is the canonical isomorphism(Am, Im, m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order pm for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that pm m = . Proof. If (b) is false for a given m, then there exists an element G, of order pr for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (pm) G pm G). There exists an opensubgroup Go of G such that (G0) = 0 and has order pr in G/G0.Choose H to be the subgroup of G generated by , and then aneasy application to G/G0 of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k1 F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'ab/k2) whose image $$\stackrel{\¯}{\sigma}$$ in Gal (k1/k2) generates this last group. The error occursin the statement that the canonical map v : AP A acts on pointsby sending ap a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that p = 1 (in Gal (F'/k2)). By applying Lemma 2 to G = Gal (F'ab/k2) and the map G Gal(k1/k2) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that pm = 1, for some m, andG = G' x H where G' acts trivially on k1 and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k1 k2is a cyclic Galois extension of degree pm. In the first case, we let K F'ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k2. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\¯}{\sigma}$$, I$$\stackrel{\¯}{\sigma }$$, $$\stackrel{\¯}{\sigma}$$) be an isomorphism defined over k1 and let v ... p–1 = µ(R). If is replaced by for some Autk1((A, I, )) then is replacedby P. Thus, as µ(R) is finite, we may assume that pm–1= 1 for some m. Choose K, as in (b), to be of degree pm overk2. Let m be a generator of Gal (K/k2) whose restriction tok1 is $$\stackrel{\¯}{\sigma }$$. Then : (A, I, ) (A$$\stackrel{\¯}{\sigma }$$, I$$\stackrel{\¯}{\sigma}$$, $$\stackrel{\¯}{\sigma }$$ = (A$$\stackrel{\¯}{\sigma}$$m, I$$\stackrel{\¯}{\sigma }$$m, $$\stackrel{\¯}{\sigma}$$m is an isomorphism defined over K and v mpm–1, ... m =pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk2 which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of R*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).  相似文献   

4.
Let be an infinite cardinal and let G = 2. Now let β Gbe the Stone–ech compactification of G as a discrete semigroup,and let =<cβ G {xG\{0}:minsupp (x)}. We show that thesemigroup contains no nontrivial finite group.  相似文献   

5.
In this paper, the behaviour of the positive eigenfunction of in u| = 0, p > 1, isstudied near its critical points. Under some convexity and symmetryassumptions on , is seen to have a unique critical point atx = 0; also, the behaviour of both and is determined nearby.Positive solutions u to some general problems –pu = f(u)in , u| = 0, are also considered, with some convexity restrictionson u. 2000 Mathematics Subject Classification 35B05 (primary),35J65, 35J70 (secondary).  相似文献   

6.
In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants ae(), b() and a certain cardinal invariant md(+)on +. As a corollary we get for example the following result.For a successor cardinal , even assuming that < = and 2= +, the following is not provable in Zermelo–Fraenkelset theory. There is a +-cc poset which does not collapse andwhich forces a() = + < ae() = ++ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary).  相似文献   

7.
Let be a projective unitary representation of a countable groupG on a separable Hilbert space H. If the set B of Bessel vectorsfor is dense in H, then for any vector x H the analysis operatorx makes sense as a densely defined operator from B to 2(G)-space.Two vectors x and y are called -orthogonal if the range spacesof x and y are orthogonal, and they are -weakly equivalent ifthe closures of the ranges of x and y are the same. These propertiesare characterized in terms of the commutant of the representation.It is proved that a natural geometric invariant (the orthogonalityindex) of the representation agrees with the cyclic multiplicityof the commutant of (G). These results are then applied to Gaborsystems. A sample result is an alternate proof of the knowntheorem that a Gabor sequence is complete in L2d) ifand only if the corresponding adjoint Gabor sequence is 2-linearlyindependent. Some other applications are also discussed.  相似文献   

8.
Packing, Tiling, Orthogonality and Completeness   总被引:3,自引:0,他引:3  
Let Rd be an open set of measure 1. An open set DRd is calleda ‘tight orthogonal packing region’ for if DDdoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of Rd. The main contribution of this paperis a new way of proving the following result: D tiles Rd whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L2(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in Rd, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ Rd withtranslates of a certain non-negative function and, additionally,we have completeness of E in L2() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.  相似文献   

9.
The statement of the numerical values and z0 on page 167 of[1, Section 3] contained an error. The values that were actuallyused were (to nine decimal places): thesebeing shifted, by the periods 21 and 23 respectively, comparedwith the values given in [1] (with 1 = 1.496729323 and 3 = 1.225694691i).With 0 = 1 and (z) denoting the sigma function (z; g2, g3) withinvariants g2 = 4, g3 = –1 associated with the ellipticcurve given by equation (3.2), these values of and z0 yield and the latter three values all agreewith those stated in the paper (apart from rounding down thelast digit in the imaginary part of A). 2000 Mathematics SubjectClassification 11B37 (primary), 33E05, 37J35 (secondary).  相似文献   

10.
In this paper, the space A (D)is considered, consisting of thoseholomorphic functions f on the unit disk D such that || f ||= supz D | f(z)|(|z|) < +, with (1) = 0. The sampling problemis studied for weights satisfying ln (r)/ln(1 – r) 0.Möbius stability of sampling is shown to fail in this space.2000 Mathematics Subject Classification 30H05 (primary), 30D60(secondary).  相似文献   

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