Decomposition of Positive Sesquilinear Forms and the Central Decomposition of Gauge-Invariant Quasi-Free States on the Weyl-Algebra Reinhard Honegger
Decomposition of Positive Sesquilinear Forms and the Central Decomposition of Gauge-Invariant Quasi-Free States on the Weyl-Algebra Reinhard Honegger
Institut für Theoretische Physik, Universität Tübingen
Z. Naturforsch. 45a, 17-28 (1990); received June 10, 1989 A decomposition theory for positive sesquilinear forms densely defined in Hilbert spaces is developed. On decomposing such a form into its closable and singular part and using Bochner's theorem it is possible to derive the central decomposition of the associated gauge-invariant quasi- free state on the boson C*-Weyl algebra. The appearance of a classical field part of the boson system is studied in detail in the GNS-representation and shown to correspond to the so-called singular subspace of a natural enlargement of the one-boson testfunction space. In the example of Bose-Ein- stein condensation a non-trivial central decomposition (or equivalently a non-trivial classical field part) is directly related to the occurrence of the condensation phenomenon.
Key words: Closable and singular positive sesquilinear forms; gauge-invariant quasi-free states on the Weyl algebra; central decomposition; classical part of boson fields; condensation phenomena.
1. Introduction sesquilinear form in the limiting Gibbs state of the infinite Bose system. From the very idea of Bose-Einstein condensation, In the present work we generalize the above situ- that the zero momentum is macroscopically occupied, ation. First we investigate the decomposition of an it follows that the thermodynamic equilibrium distri- arbitrary positive sesquilinear form t (defined on a bution should have some kind of singular behaviour. pre-Hilbert space E) into its closable and singular This was in fact confirmed by rigorous investigations part. Then we use this decomposition of t to deduce of various forms of the Bose-Einstein condensation, the central decomposition of the gauge-invariant where the singularity only appears in the infinite quasi-free state co on the Weyl algebra over E associ- volume limit with fixed particle density, which should ated with the form t. There is a one-to-one correspon- be larger than some critical value. In [1], [2] and [3] (see dence between the appearance of a non-trivial singu- also the fundamental work in this field [4]), in the lar part of t and a non-trivial central decomposition of formalism of operator algebraic quantum statistical co. Thus condensation appears if and only if the central mechanics for this kind of thermodynamic limit the decomposition becomes non-trivial, and the latter is limiting Gibbs states have been deduced, each of only possible for infinite boson systems. In the usual which is gauge-invariant and quasi-free and hence statistical interpretation, the central measure gives the uniquely determined by a positive sesquilinear form classical-statistical mixture of the ensemble into dis- defined on the one-boson testfunction space. Below joint primary (purely quantum mechanical) states. the critical density these sesquilinear forms are clos- The disintegration via the central measure (spatial able, whereas above the critical density there is an decomposition) of the weak closure of the GNS-repre- additional singular sesquilinear form, which is given by sented Weyl algebra appears in the form of a tensor the evaluation of the testfunctions at zero momentum product, one factor of which being a factorial von and thus yields the non-closability of the total positive Neumann algebra and the other one being merely form. Hence the occurrence of condensation is closely commutative. Hence in the reconstructed quantum related to the occurrence of a non-closable positive mechanics (GNS-representation of a>) the mentioned classical-statistical mixture - which also represents the collective phenomenon of condensation - is expressed Reprint requests to R. Honegger, Institut für Theoretische by an additional classical field. The latter arises from Physik der Universität Tübingen, Auf der Morgenstelle 14, D-7400 Tübingen, West-Germany. the singular part of t, whereas the purely quantum
0932-0784 / 90 / 0100-0017 $ 01.30/0. - Please order a reprint rather than making your own copy. 18 R. Honegger • Decomposition of Positive Sesquilinear Forms mechanical one (the factorial algebra) is related to the developments of Sect. 2 are indicated to treat the clas- closable part of t. In the above case of Bose-Einstein sical states with finitely many (eventually unbounded) condensation the classical phase space is two dimen- modes. The general class of the classical states is de- sional and can be parametrized in terms of modified fined in terms of an operator algebraic generalization polar coordinates, which are interpreted as the (scaled) of Gauber's P-representation with positive measures particle density of the condensate, R, and a phase angle, and includes the coherent and gauge-invariant quasi- which is in connection with the gauge transforma- free states. For the classical states the really macro- tions. Due to the statistical interpretation, each (infinite) scopic (classical) modes may be identified by searching boson system of the ensemble has sharp values R and for the singular subspace. In the frame of classical
3 and is described by the state coR 9 appearing in the states it seems to be possible to study in a rigorous oo 2k way some classical non-linear dynamical equations of central decomposition of at: a> = J j coR 9 d/x(R, $). R=0S=0 quantum optics, which now are basing on a complete The classical features of the boson field also can be microscopic theory and not on the usual heuristic seen on the C*-algebraic level. The decomposition arguments. These dynamical equations one may get theory of positive sesquilinear forms (cf. Sect. 2) from microscpic quantum interactions by the above allows an enlargement of the one-boson testfunction procedure, a suitable enlargement of the one-boson space E, to a Hilbert space JT and consequently sug- testfunction space and a subsequent separation of the gests also a natural enlargement of the C*-Weyl alge- global classical dynamics by letting go the testfunctions bra (which now is defined by means of a degenerate into the singular subspace. The indicated approach to symplectic form on Jf), so that the classical field part this field of quantum optics would have the advantage may vary independently of the quantum mechanical of studying both the quantum mechanical and the one. The enlargement of the algebra is adapted to at, classical aspects of such dynamics and their dynamical and thus co may be extended to this new C*-Weyl interaction. (For references see the citations in [6].) algebra. Due to the orthogonal decomposition of Jf After this outlook we return to the present investi- into its closable and singular subspaces (see Sect. 2) gations and proceed as follows. Section 2 is devoted to the new Weyl algebra divides into a tensor product, a detailed analysis of the above mentioned decompo- and so does the extended a>. In this way the C*-boson sition of a positive sesquilinear form t defined on a system divides tensorially into a quantum mechanical pre-Hilbert space E. In contrast to closability of a C*-system and a classical (commutative) one, whose positive form (which then arises from a positive self- GNS-represented weak closures agree with the above adjoint operator), we introduce the notion for a form mentioned factorial and commutative von Neumann to be singular. The basic idea of the decomposition algebras. Now the field operators associated with the of t, as given in [7], is the completion of £ with respect GNS-representation of the enlarged Weyl algebra are to a stronger hilbertian norm, namely the sum of the well defined, and the merely quantum mechanical and old one and the quadratic form, followed by an the merely classical field expressions are approx- orthogonal decomposition of this completion into the imate by letting go the testfunctions in the argument so-called closable and singular subspaces. In the stan- of the (smeared) field operators from E into the clos- dard literature on quadratic forms such completions able or singular subspace of Jf, respectively. are usually only considered for closable forms. In the A similar procedure is also possible for the class of context of real symplectic spaces similar completions coherent states. We refer to [5], where the macroscopic and decompositions are done in [8]; they agree with (classical) aspects of coherent light with a high photon those here if the imaginary part of the inner product density has been discussed. of £ is identified with the symplectic form. But because As pointed out above, the concept of the decompo- E is an inner product space (and hence closability and sition of positive sesquilinear forms with the closable singularity of forms are naturally defined in terms of and singular subspaces is essential for the study of the inner product topology), the structure of the de- both the purely quantum mechanical and the classical composition theory becomes here much richer than in parts of boson fields within the classes of the gauge- the symplectic case. Also a comparison of positive invariant quasi-free states and the coherent states. forms is introduced, which among other results shows One may ask, wether this concept also applies to other that the above decomposition of t is the one with the classes of states. Indeed, in [6] methods basing on the largest closable part and the smallest possible singular 19 R. Honegger • Decomposition of Positive Sesquilinear Forms one, and is unique in this sense. Finally, for applica- define a new scalar product by tions of the theory some examples of decompositions £ x £ —• IK, (/, g)
Using the decomposition of t from Sect. 2, the GNS- X and (Uf\Ug)x = E, which lifts to a regular Borel measure on the state Since by (2.1) one has || Vf\\x> \\f\\ V/e E, the space of #"(£) decomposing co. The latter is shown to map U'1: U(E)-+E, U{f)\-*f has a contractive be the central measure of co (cf. also [8]). Doing this extension F:Jf->Jf. Then Jfs := {>/ e X | Vrj = 0} requires more than showing the mutual disjointness of = ker(F) is a closed subspace of Jf, called the singular the supporting states [9] and is achieved by an ex- subspace (corresponding to t). plicite calculation of the Tomita map. In the case of a finite dimensional singular part of t by means of Lemma 2.2. For the positive form t the following condi- Fourier transformation the central measure is con- tions are equivalent: structed explicitly and applied to the above mentioned (i) t is closable; limiting Gibbs state of the Bose-Einstein condensation. (ii) if(f„)neN is any sequence in E so that lim || /„ || = 0 n -»oo and(Uf„)net 2. Decomposition of Positive Sesquilinear Forms Proof: Transferring by U the notion of closability to Jf gives (i) <=> (ii). Let Jf be a fixed Hilbert space over K with scalar (ii) => (iii): Let rj e Jfs and (/„)neN a sequence in E product <.,.) and corresponding norm || • ||, where IK with lim || Ufn — rj 11^ = 0. From the continuity of V n oo denotes the real field IR or the complex field Lemma 2.3. For the positive form t the following condi- Let feE and (/„)„gN be a sequence in E such that tions are equivalent: lim Ufn = PcUf. Then lim /„ = lim Ft//„ = FPCUf n -»oo n-»x n -»oo (i) If t' is a closable positive form with t' (v) for each f e Jf there exists a sequence (/„)neN in E Definition 2.5. The form tc is called the clasable part, with lim || /„ —/ ||=0 and lim t(fn,fn) = 0. and ts is called the singular part of the positive form t. n -»oo n -» oo To the closure ^([10], Theorem VI.1.17) of the clos- Proof: (i) => (ii): If t = t + t , then t = 11 WII^-ll/ll2=o. and (iv) => (i): Assume to be a closable positive lim tc (/„ —fm, /„ -fm) = 0, form with f'^f. By (iv) for feE let (/„)„ em be a from which /e 2(tc) follows. Conversely, let /e 3>(tc) sequence with lim /„ = / and lim t(fn,fn) = 0. Hence n -* oo n -»oo and (/„)neN a sequence in £ with lim ||/„ —/1| = 0 and n oo lim t{fn -fm ,f„-fm) = 0 and because of r' < t one n. m oo lim W„-/,/„-/) = 0. Then (Pc £//„)„ gN is Cauchy in has lim t'(f -f ,f„-f ) = 0. Since t' is closable we n -* oo n, m -»oo n m m Jff with limit £ e Hence F£ = lim VPcUfn = lim /„ compute 0 < t'(f,f) = lim t'(fn, fn) < lim t(/„, /„) = 0. n -» oo n -> oo =/ The rest follows from (T 2fT 2g) + (fg} = (iv) <=> (v) is obvious. • - c c te(f, 9) + a singular positive form ts. We remark that the decom- Theorem 2.4. The form tc is closable and ts is singular. position of a positive form into a closable and a singu- lar positive form is not unique, which is seen from the Proof: If (/„)„ is anY sequence in E with eN following simple example: Let [a, b] be a finite closed lim ||/„ ||=0 and lim t (f -f , f -f ) = 0, then c n m n m interval of the real line and Jf = L2([a, b]) and n -»oo n, m oo b _ (P Uf„) is Cauchy in Jf converging to some vector c nehl h (/.0):=J/Vd* and t] e Jfc. Consequently Vrj= n li-»m oo VPcUfn = n li-»m oc VUfn a = lim /„ = 0. Hence >/ = 0 since F is injective on Jf^. n -* oo t2u; 0) := /(c) .9(c) for some c e [a, b], 2 Thus lim tc(f„,f„)= lim (|| Ufn — || f„ || ) = 0 and n -» oo n oo where ^(t1) = ^(t2) is the set of all feJf such that / r„ is closable. is absolutely continuous on [a, b] with derivative f'e 21 R. Honegger • Decomposition of Positive Sesquilinear Forms It is well known that tx is closed. t2 is singular (see Proposition 2.9. Let t' be a singular form satisfying Proposition 2.11 below). But t2 is relatively bounded ts II/II < || U'f\\x.<\\ Uf\\x V/e E, there is a selfadjoint Lemma 2.7. If t' is any closable form satisfying t' < t, operator A on X with then if follows t' Lemma 2.8. In the above situation Ö(XS) c I(XS'). Using (2.5) and (2.6) we get Ps = PS(C — A) Ps = Ps Ö p; Ö Ps = Ps p; Ps, from which we get Proof: Let r] e Xs and (/„)neN a sequence in E with lim || Ufn - rj || ^ = 0. Then 0=Vrj= lim VUfn = lim /„. ps=psp; = p;ps. (2.8) B-»OO noo B-»OO 1 (IV) Let t=t'. Since + *(/,g) = «f,g> + Now, if we put rj':= I ~ (Ö r]) we obtain 0 = 2 + tc(f, g)) + «f, g> + ts(f, g))- Proof of Lemma 2.7: Since t' is closable we have Hence t (f f) — t'{f f). • by Lemma 2.2: Jfs' = {0}. Hence by Lemma 2.8: c c ker(d). From (2.5) one gets ker(d) = ^. Con- We now turn to some examples which are based on sequently PCÖ =ÖPC = Ö, from which \\U'f\\x.= two measures mutually singular to each other. By a II <3 uf \\x = II ÖPC Uf \\x < II Pc Uf \\x follows. • unique lifting from an L2-space to a function space E we mean that in each L2-equivalence class there is a most Also there is a useful result for determining the one element of E. decompostion t = tc + ts: Lemma 2.10. Let t = t' +1", where t' and t" are arbitrary Proposition 2.11. Let X be a set, © a a-algebra on X and H and v two measures on 53 singular to each other. Let positive forms such that ts <.|.)r, where (f\g\. := + f(f, g) for all (that is, singular with respect to the scalar product of fgeE. L2(X,p)). 2 2 Remark: If t' is closable, the condition ts sequence (/„)„eN in E such that the kernel ker(F) = Jfs = {/eJT|/(X) = 0 VxeX\Y} and lim t"(/„,/„) = 0. Consequently (C//J N is 2 n-* oo which is isomorphic to L (X, v). Cauchy in jf, converging to an rj e jf. Since fs < t", Decompositions into the closable and singular parts we have on L2-spaces one can obtain by multiplication opera- tors and singular measures. We state an example 0= lim t (f„,f„)= lim || P UfJ2 = || P rj H2,. s s x s relevant for the Bose-Einstein condensation. 23 R. Honegger • Decomposition of Positive Sesquilinear Forms Example 2.12. Let E be the Fourier transform of the set which is dense in if M is finite dimensional. For a of all infinitely differentiable (£,-valued functions on Rm regular state co one can introduce annihilation and with compact support and v a finite Borel measure singu- creation operators lar to the Lebesgue measure A on Rm. Further let 2 m (p e L (R , A) and M C, (f g) i—• jRm fg dv is the singular part of the <£(/):= ~7= (®c> (/) - * & 2 /e E. An equivalent condition for a state co to be regu- that for each f e E there is a öf > 0 such that lar is the continuity of t e IR i—> co (W(tf)) for each / e E. oo i" t" The infinitesimal generators - the field operators - of < where all the series converge for t e] — öf, öf[. 24 R. Honegger • Decomposition of Positive Sesquilinear Forms An analytic state co is called quasi-free if Hence by the Lemma 3.1 each gauge-invariant quasi- cM*a>(/i);.--; t(f,g) = 2co(a*(g)aJf)) = 2coT(a*(gy,aJf)) (3.4) Lemma 3.1. Let co e y. The following conditions are (t is positive since r(/,/)= || ajf) ||2 > 0). Con- equivalent: versely, it is well known in the literature [15], [8] that (i) co is quasi-free; for each positive form t on £ there exists a unique (ii) co is analytic and a> {$ (f)n) = 0 for all n > 2 and T V) gauge-invariant quasi-free co e Sf satisfying (3.3) (com- all /£ £; pare also the remark in Subsection 3.2). (iii) for each feE there is a polynomial P : 1R ->(E f If t = 0, then (3.3) determines the Fock state co^ e y. such that a>{W(tf)) = exp{P (t)} VfeR. f Its GNS-representation is given by the Bose-Fock If one of these conditions is fulfilled, co is entire analytic space tF+(E) over £, the vacuum vector Q^ = (1,0,0,...) and the usual Fock representation n#{W(f)) = W^{f) andP (t) = itco( = J^^) (g) &+(r2), where ^ =(1 + T)>(£) and 2 co(W(tf)) = exp ji t coT( + X coT( Lemma 3.2. It is (77 , jV , QJ = (77 L- >> ß») and from which (i) follows. The proof of (iii) => (i) is due to w a ö (£ Ji^—Mö). [16], we briefly refer: Since co is a state, the function 1R 3 t (—• co(W(tf)) is positive-definite. However, such Proof: Using the Weyl relations (3.1) one gets for 0 that exp Pf is positive-definite, the only possibility for f,g,he 3>{t) the polynomial Pf is to have a degree less or equal two. II {n*mf))-n*(W(g))) n*mh)) Oa II2 Now compare with (3.2). • = cö(W(-h)[W(-f)-W(-g)] [W(f)~ W(g)] W(h)) A state co on #"(£) is called gauge-invariant, if = 2-2 ^(e'^^/'^ + ^M-/))) co(W(ei9f)) = co(W{f)) for all 9 e R and all fe E. If co is analytic, using (3.2) and the CCR, the gauge-invari- •exp{-i(||/-0||2-F(/-0,/-0))}. ance of co is equivalent to co(a*(/)m a (/)") = 0 for all (O Thus, if Q>(t) is equipped with the norm ||/||2:= m + n and all feE. \\f\\2 + t(f,f) (compare (2.1)), the map @{t)sf Using the CCR one gets i—• 77,0(W{f)) is continuous in the strong operator to- 2 L 2 2 2 ^Jf) = 2(a*Jf) +aJf) + 2aZ(f)aJf)+\\f\\ t). pology. • 25 R. Honegger • Decomposition of Positive Sesquilinear Forms 3.2. The Central Decomposition Lemma 3.3. The measure is regular. of Gauge-invariant Quasi-free States Proof: Let B be a Borel subset of Sf. Since p. is regular In this subsection let co be a fixed gauge-invariant and q^iK) is compact, if K is compact, we get quasi-free state on the Weyl algebra W(E) determined pJB) = p(q~1(B)) by (3.3) with the positive form t: ExE -> (C. As in Section 2 let U: E Jf be the injection of E into its = sup{p(K) I K c q~1 (B), K compact} completion with the scalar product (2.1) and V: JT-> E = sup{pJqJK)) | qJK) <= B, K compact} the continuous extension of and t = tc + ts the decomposition of t into its closable and singular part < sup {pw(K)\ K c B, K compact} with closable subspace Jfc and singular subspace Jfs. Denote by coc the gauge-invariant quasi-free state de- That is: p^ is inner regular. Now let Ac :=Sf\A. Since fined by (3.3) with the form tc and coc its canonical £f is compact, each closed subset K ^ £f is compact. extension (3.5). From the inner regularity of p we get If we regard E as an additive group equipped with m the discrete topology, the associated character group pjautomorphism xx on iV(E) so that c tx(W{f)) = x(f) W(f) V/e£. If Jfs is the character pJB) = pJ^)-pJB ) group of the discrete group pTs, -l-), each lifts to a := e = inf {pJ Because lim Xi = X in the -topology of is equiva- (o(W(f)) = a>cmf))™V(-k\\PsUf\&} = J co °x .(W(f))dp(x) lent to lim &(£) = *(£) e Jfs ([17], (23.15)), it follows c x ie I we arrive at the integral decomposition of our gauge- lim qJXi) (W(f)) = lim Zi(Ps Uf) coc(W(f)) iel ie! invariant quasi-free state co e = X(PS uf) coc(W(f)) = qJx)(W(f)), co = j coc ° xx. dp{x) = j (P dpj(p). (3.8) n e which gives lim q(0(Xi) = q<0(x) i th weak*-topology, ie/ Remark: Using similar arguments one can prove showing qm to be continuous and Nw to be compact since Jf^ is so. that indeed each positive form t: E x E -* C by Because 3 £ h-> exp {— | || is positive-defi- (3.3) defines a gauge-invariant quasi-free state on nite, by Bochner's theorem ([17], (33.3)) there is a exp{ — ht(ff)} is positive-definite + unique measure p e A/ + (jQ (the finite positive regular and determines a measure q e M {E) which lifts to a Borel measures on JQ, such that regular measure q, on Sf via the map pt(x) := co^ ° ix. Thus co := j (pdgt((p) defines a state on iV(E) satisfy- exp{-i H\\lr} = I x(t)dn(x) V£eJfs. (3.6) ing (3.3). * Identifying the elements of L°°(jrs, p) with multipli- Let us transfer // to a Borel measure p^ on by cation operators on \J(Xs,p) and introducing for setting each r] e Jfs the function pJB) := p(q~1 (ß)) for each Borel set B c (3.7) F„:Jfs^C, x^X(rj) As a consequence Hto(£f\Nm) = 0 and the spaces we get a result, which is a slight generalization of one W(Xs,p) and LP («9", pj) can be identified for each in [4] and analogous to one in [18], [8]. Observing p e [1, oo]. Lemma 3.2 and Proposition 2.6 we have with V= VPC: 26 R. Honegger • Decomposition of Positive Sesquilinear Forms Proposition 3.4. Let WJQ := nw(W{V£)) ® FPs Similar to (3.5) it is possible to extend co to a larger C e Jf. Then the GNS-representation of co is given by Weyl algebra. For doing this we introduce on Jf the symplectic form o>(, fgeE continuously to njw(f))=wjuf) V/e E JT x JT. By Proposition 2.6 Jfs = {£ e JT | ^) = 0 V^eJf}. Then co extends to a state cö on the with i(z)=l- Moreover, Jt = Ji ® L® (jt , /i) and m (j)c s Weyl algebra A (Jf, a(see the appendix), satisfying f/ie map W^.y. Jf is continuous with respect to cö(<5^) = exp{ — \ || ^ V£eJT. Moreover, by Pro- the norm of Jf and r/ie strong operator topology of JtUJ. position 2.6 and Lemma A. 1 we have A(JT,ar) = Proof: The continuity of Ww (.) follows from the proof A(JTc,ax) ® J(JQ £ W~(®{TC)) ®V(jrs) and cö be- of Lemma 3.2 and from a similar argument for the comes the product state cö = Wc ® p, where p is from state on jts), regarded as multiplication operators, and = WJQ, £ e Jf and = and Q^ = Qa. observing that LH {FJ rj e J^} is dense in by If (£,<.,.» is separable, each state (p = ooc° xx> the Stone-Weierstraß theorem. Now, if £ e Jf, then occurring in the support of the measure pm has the lim || £//„ — £ 11^=0 for some/„ e E and s- lim Ww(Ufn) GNS-representation (77^, , ) with separable n -»oo n -» oo ^ and n¥(W(f)) = z(P.Uf)na}jV(f)). Moreover = W^(ft_Hence, WJi)^® F„e*a for one easily checks the measurability of the von Neu- and = for fe®(tc). mann algebra field (p ^Jt^ ([20], Chapitre II § 3). Use [19], Theorem III-1.2, and the rest is obvious. • Hence the direct integral exists and by Effros' theorem Thus, since Jtm is a factor, the center of Jtw is given ([13], Theorem 4.4.9) we get the spatial decomposition by % = Jt nJt' = l ® L00 (Jf , /v) . (3.9) (na,jra,Q„)= j (n^jf^QjdpM, (0 01 (0 Ue s No, 00 ® Identifying L°°(Jts, p) and L pj (via the map A> = f PJ = J Fn(x)0)c°Tx.mf))dp(x), and supports p. As in Lemma 3.3 one proves regular- ity of p and hence p e M + (R2") is the unique measure 2 from which with (3.10) one gets xß(Fri) = WV){fi) satisfying (3.6). Similarly one checks gc,J(C(R ")) ^ N„ = tCOc ® Fn. xli(F) = tWc ® F follows from the linear- to be Borel and supporting pw. Remembering x ity and the o(L , L^-continuity of xß. Hence xß is a C(x)' = C(x) ° Ps o U e E, additional to (3.8) we have x *-isomorphism from L (Xs, p) onto and pw is cen- co= f C0c° TC(xydg(x). (3.11) tral by [13], Proposition 4.1.22. • R2" R. Honegger • Decomposition of Positive Sesquilinear Forms 27 One may ask if there is a similar structure also for A (H, a) be the complex vector space generated by the infinite dimensional This is the case if PSU(E) is a functions öx (x e H) from H to C defined by nuclear space and ||. || # a continuous hilbertian norm on PSU(E). By the Bochner-Minlos theorem one gets a Gauß measure Q on the dual PSU(E)*, whose canonical image measure in Ps U (E) agrees with A (H, o) is a *-algebra with unit <5 with respect to the + 0 p e M (P^J(E)) defined similar to (3.6). a{x y) product öx • öy = e~2 ' and the involution Now let us turn to the example of Bose-Einstein <5* = <5 _JC. In [21], by means of the completion with the condensation: E is the space of all Lebesgue square minimal regular norm the C*-Weyl algebra A (H, a) is integrable functions on IRm with compact support. obtained. If H is a pre-Hilbert space and (j{x,y) The limiting Gibbs state a/ above the critical particle = 3 Example 2.12 with the Dirac measure v = y<50. Hence, : 2 g) = (Ty, T g) is the closable part and Lemma A.l. If a = 0 and H is the compact character ts(f, g) :=yf(0) <7(0) = y \Jg d<50 is the singular part of group (A-topology [17], (23.13)) of the additive discrete 2 m t and JTS ^ L (lR , y<50) = C with PJ=]/yf(0). Set- group (H, +), then there is a unique *-isomorphism y ting /(0) ^ (9?7(0), 3/(0)), from (3.11) one obtains the from the abelian C*-algebra A(H, 0) =: A(H) onto the central decomposition of coß continuous functions H) of H so that y(öx) = Fx x e ß ß Vx e H, where Fx(x) '•= x( ) VX H- co (W(f)) = aj (W(f))~ J ei]/y [1] J. T. Cannon, Commun. math. 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