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A period August 2 4 0 , 1 20 0 1period 0 ; Published 0 72 online September 1 6 , 20 1 0 Abstract period The integral in noncommutative geometry open parenthesis NCG closing parenthesis involves a non hyphen st andard trace nnoindentd o i : 1$ 0 .nddagger 3 8 42 / SI$ G School M A . of 2 0 Mathematics 1 0 . 0 72 and Statistics , n h f i l l University of New South Wales , Sydney , 2052 , Australia called .. aAbstract Dixmier .. traceThe periodintegral .. in The noncommutative geometric origins geometry .. of this ( NCG integral ) involves are well a knownnon - st period andard .. trace From a measure hyphen theoretic view comma .. however comma .. the formulation contains several difficulties period .. We re hyphen n centerlinecalledfE − amail Dixmier : nquad tracef . The s n geometricquad u k originsnquad ofo this ch integralnquad aree v well $ known @ $ . u nFrom s nquad w . edu . nquad a u g view resultsa measure concerning - theoretic the technical view , features however of , the the integral formulation in NCG contains and some several outstanding difficulties . We re - problems in this area period The review is aimed for the general user of NCG period nnoindent Receivedview results March concerning 25 , the 20 technical 1 0 , features in final of the form integral August in NCG 4 and , 20 some 1 0 outstanding ; Published online September 1 6 , 20 1 0 Key words : .. Dixmierproblems trace semicolon in this area .. singular . The review trace is semicolon aimed for .. the noncommutative general user of integration NCG . semicolon .. noncommutative geometry semicolon Lebesgue integral semicolon noncommutative residue nnoindentKeydoi words : :10.3842/SIDixmier trace ; singularnquad traceGM ; noncommutativenquad A . 2 0 integration 1 0 . 0 ; 72 noncommutative 201 0 Mathematics Subject Classificationgeometry ; Lebesgue : 46 L 5 1 integral semicolon ; noncommutative 47 B 1 0 semicolon residue 58 B 34 Contents n centerline f Abstract201 0. Mathematics The integral Subject in Classification noncommutative : 46 L geometry 5 1 ; 47 B 1 ( 0 NCG; 58 B ) 34 involves a non − st andard trace g 1 ....Contents I nt ro duction .... 2 2 .... Dix .... m i e r .... t r a c e s .... 5 n centerline1f c a l l e d nquad a Dixmier nquad I ntt r ro a c duction e . nquad The geometric origins nquad of 2 this integral are well known . nquad From g 2 period2 1 .. 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C l o s e d s y m .. m e tr i c .. s u b id e als .. period .. period .. period .. period .. period .. period .. period .. period .. n centerline......................14fKey words : nquad Dixmier trace ; nquad singular trace ; nquad noncommutative integration ; nquad noncommutative g period4 .. period Origin .. period of .. period t .. period h e .. period no .. n period c o .. period m .. m period u .. t periodat i v ..e i period n t egr .. period al .. 15 period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 4 n centerline4 .f 1geometry N o r m ; al Lebesgue e x te ns ion integral . ; . noncommutative . residue . .g . 4 ...........................17 Origin .... of .... t .... h e .... no .... n c o .... m .... m u .... t at i v e i n t egr al .... 15 4 period 1 .. N o r m .. al e x te ns ion .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. n centerline4 . 2f201 T o 0ols Mathematics fo r t h e n o nSubject co m Classification m u t a t iv e i n t: e 46 g ra L l 5 . 1 ; . 47 . B . 1 0. ; . 58 . B . 34 .g period....................19 .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. nnoindent5Contents Z eta f u n c t i o n s a n d h e a t k e r n e l s 20 period .. 1 75 . 1 R esi d u e s o f z e ta fu n c t io n s . 4 period 2 .. T o ols fo r t h e .. n o n co .. m .. m u .. t a t iv e i n t e g ra l .. period .. period .. period .. period .. period .. period .. nnoindent.......................201 n h f i l l I nt ro duction n h f i l l 2 period .. period5 . 2 .. period H e at .. k perioder n el .. asperiod y m .. periodp to t i .. cs period . ... period . .. .period . .. period . .. period . .. period . .. period . .. period .. period .. period .. period .. period .. period .. period .. period .. period .. period .. 1 9 nnoindent......................232 n h f i l l Dix n h f i l l m i e r n h f i l l t r a c e s n h f i l l 5 5 ....6 Z eta C fh .... a rac u n te.... r c i t s i oi n .... ng s t .... h ae .... n do .... nh ec a o t .... m k e r m .... n ue l tative s .... 20 i n te g r al 24 5 period 1 .. R .. esi d u .. e s o f .. z e ta fu n c t io .. n s .. period .. period .. period .. period .. period .. period .. period .. period .. n hspace ∗fn6 . 1f i l lChg2 a r. a 1 n cquad t e r isS a in ti on gu a n l a d r sn inquad gu las r y t m r anquad c e smetri . c . f . un . c .t ion . als .nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad . nquad 1 0 period....................24 .