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Additional resources for An Introduction to Multicomplex Spaces and Functions
Let N, N be two multi-matrix subalgebras of a multi-matrix M M algebra M, given together with an isomorphism cp: N --; -N. If AN = A_, then cp N extends to an automorphism 0 of M. 3. Inclusion matrix and Bratteli diagram Proof. We show this for M a factor, and the general case will follow; we may thus identify M with Endl«V) for some vector space V over I<. Set M M A N =A_=(A 1,···,A )EM 1 (IN). N j=1 J Ell be decompositions into factors, where the indices are chosen so that cp(qj) = qj for j = 1,.
Secondly, a more interesting variation comes from using tensor products M ®N M ®N . (a Tor-like idea) instead of endomorphisms ... Endtr(EndN(M)) (an Ext-like idea); we refer to [Jo4] for this. One could check that these variations give the same index for semi-tlimple pairs, but more general examples may have several indices. The connection of this subject with Chapter 1 is this: For inclusions 1 E N c M of semi-tlimple algebras over a perfect field, the index turns out to be the square of the norm of a certain matrix of natural numbers A~ associated with the pair of algebras.
A path ( ... ,ek) has end equal to end ( ek); a path (eO'···) has start equal to start ( eO). If e and TJ are paths such that endW = start (TJ) we define eOTJ to be the e, then TJ". path "first = cp(start( ek)) ° A path :8 is monotone increasing if cp( end( ek)) e on + 1 for all k. We let denote the set of infinite monotone increasing paths on :8 starting at *j O[r the set of infinite monotone increasing paths starting on the rth floor of :8; Or] the * set of monotone increasing paths starting at and ending on the rth floor; and O[r,s] the set of monotone increasing paths starting on the rth floor and ending on the sth floor (r< s).
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