4 EDWARD G . EFFRO S either H(B) = {0 } o r it i s a closed subspac e o f infinit e dimensio n i n th e "usual " discret e basis sense, it i s natural t o introduc e th e continuous dimensio n di m H(B) = n(B) wher e n is Lebesgu e measure. Thi s real numbe r clearl y reflect s th e relativ e "size " of suc h a space. Furthermore i f on e restrict s t o appropriat e isometries , viz., to thos e induce d b y measur e isomorphisms o f th e correspondin g subsets , two subspace s have th e sam e dimensio n i f an d only i f the y ar e isometric . The general theor y o f continuou s dimension s wa s established b y Murra y an d vo n Neu- mann nearl y fift y year s ag o [50] . Replacin g close d subspace s b y th e correspondin g opera - tors, i.e., by th e orthogona l projection s o n thos e subspaces , they wer e led to stud y th e se t of projection s i n a "rin g o f operators" . Th e latter, no w referred t o a s a von Neumann alge- bra, may b e though t o f a s the measure-theoretic generalization of a semisimple algebra. I n particular i f R i s a factor, i.e. , the cente r o f R contain s onl y scalars , and Pro j R i s its lattic e of projections , they showe d tha t ther e i s a natural ma p dim : Pro j R [0 , °°] suc h that fo r projections e, f E Pro j R, di m e = di m / i f an d onl y i f ther e i s a partia l isometr y v i n R mapping eH onto fH. The nee d fo r nonintegra l indice s i n geometry ha s only recentl y becom e apparent . Roughly speaking , it seem s to b e associated wit h th e desir e t o generaliz e theorem s o f th e calculus on compac t manifold s (suc h a s T) to th e locall y compac t cas e (e.g.,. R). T o illus- trate this , consider th e notio n o f windin g number . Give n a function / : T = [0 , 2ir] * —• C\{0}, w e recall tha t th e winding number of/(abou t 0 ) is given by the intege r M / ) = ^[Arg/(27r)-Arg/(0)] . Letting s p / = {n: f(n) ¥= 0}, w e note tha t N(f) i s contained i n th e grou p generated b y sp / Turning t o function s define d o n R , let u s suppose tha t /: R —• C is continuous an d bounded, an d 0 ^/(R) (ba r indicatin g closure) . I f /is periodic , e.g., it i s a linear combina - tion o f e int in £ Z) , we may defin e th e windin g numbe r of/b y regardin g i t a s a functio n on T . Thi s may b e generalized t o suitabl e aperiodi c functions/b y usin g the mean winding number N(f) = li m 4 [Ar g f(T) - Ar g / (- r ) ]. This quantit y wa s firs t introduce d b y Lagrang e i n hi s studie s o f celestia l mechanic s (se e 2, p . 138]) . If , fo r example , f(t) = a0elt + ale~i^lt + a2eiyf*f (if \a 0 \ \a x \ \a 2 1, one may regard this as superimposing epicycles on the basic cycle a0elf), thi s limit exists. I t will generally no t hav e an integral value , but rathe r wil l lie in the subgrou p Z 4- Z\/2 4 - Z\/3 of R . Mor e generally Boh r prove d tha t if/i s almos t periodic , this limit mus t li e in th e sub - group o f R generated b y s p /(the suppor t o f th e distributio n /) [6] . With th e abov e discussio n i n mind, one ca n se e how on e migh t introduc e othe r non - integral topologica l "indices " or "dimensions " into geometry . Fo r example , if M i s a non - compact manifold , sa y th e lea f o f a foliation o f a compact manifold , i t wil l generally hav e infinitely man y "holes" . On e might nonetheless b e able to formulate a notion o f the "average number of holes", considering those that occur in an expanding finite regio n (see [73] , [70]) .
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