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By Ajit Iqbal Singh

It truly is now renowned that the degree algebra $M(G)$ of a in the community compact workforce could be considered as a subalgebra of the operator algebra $B(B(L^2(G)))$ of the operator algebra $B(L^2(G))$ of the Hilbert house $L^2(G)$. during this memoir, the writer experiences the placement in hypergroups and unearths that, in most cases, the analogous map for them is neither an isometry nor a homomorphism. although, it truly is thoroughly optimistic and fully bounded in sure methods. This paintings provides the comparable common thought and distinctive examples.

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J3\ we v v v have |/(x) — / ( y ) | < £ and | f(x) — f(y)\ < s whenever kp(x*y) > 0. Therefore, v v v v v for x G K, \(fm*kp)(x) - f(x)\ < s and also \{fm*kp)(x) — f(x)\ < e. C, I / M / x J */x */x^) - / fdii\<\ JK f V JK V + | / fd(li*lip)- f / / fd(lip*n*iLp)- V JK fd(lip*ii)\ V /dM| < / |(/m*^)-/|d(^*/i)-f / JK / JK JK \}m*kp-f\dji < e(lip * ii)(K) + e»(K) = 2en(K). v This shows JK fdiifap * (i * lip) ^> fK fd/i and this completes the proof. 3. Special matrix orders on M(K) for commutative K.

We have to go one step further to prove that MR(K)+ = MRepK(K)+ = Mpd(K). 5B, D and E and Krein-Milman Theorem [40], p. 460 PDX(K) = {/ G PD(K) : 0 < /(e) < 1} is the Ll(K)closed convex cover of the set EPDi(K) = {0} U {/ G PD(K) : / is irreducible and /(e) = 1} of its extreme points in the dual L°°(K) of Ll(K). So for a AJIT IQBAL SINGH 36 \i G Ma(K), JK fdfji > 0 for / in EPDX(K) implies that fK fdfi > 0 for / in PDi(K). This gives that fK fd/j > 0 for / in PD(K). 3A and B, Ma(K)nMR(K)+ = Ma(K)nMRepK(K)+ and also the r6-closure £ ReipK of Ma(K) n M ( K ) + in M(K) is contained in M (K)+.

2 above can be closed in the weak topology coming from Mn(Cb(K)) to form M^d(K) a n d t h u s give rise t o a MATRICIALLY ORDER ADMISSIBLE PAIR n (M(K),Cb(K)). 2 Since Mn(Cb{K)) can be identified with Cb{K) this weak topology is simply n the product topology of (M(if),r 6 ) . (ii) For n G N, we equip Mn(Cb(K)) with the dual cone M^d(Ch{K)). 6. becomes a MATRICIALLY ORDER COMPATIBLE PAIR. (i) For n E N, M*(Ch(K)) = {[fjk] G Mn(Cb(K)) : J > ; *M**(/*fc) > 0 for each n-tuple (fij) in M(K)} = {[fjk] e Mn(Cb(K)) : Y,H*ii*k(fjk) >0 for each n-tuple (fij) in MF(K)} = {[fjk] e Mn(Cb(K)) : J2 l 0, for each n x m complex matrix [djr] and each ra-tuple (sr) in if, m G N} = {[/jfc] ^ M n ( C f e ( i f ) ) : [fjk(Sr * ^)](i,r)(fc,i;) € M n m + for each ra-tuple (sr) in K and m E N } .

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