# Algebraic and analytic aspects of operator algebras by I. Kaplansky

By I. Kaplansky

An algebraic prelude Continuity of automorphisms and derivations \$C^*\$-algebra axiomatics and simple effects Derivations of \$C^*\$-algebras Homogeneous \$C^*\$-algebras CCR-algebras \$W^*\$ and \$AW^*\$-algebras Miscellany Mappings holding invertible parts Nonassociativity Bibliography

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Additional info for Algebraic and analytic aspects of operator algebras

Example text

In the XXZ model, we have bk D C or bk D , for all k. In the crystal language, fC; g is a perfect crystal. 26. Let l be a positive integer, which is called a level. Let B be a finite g0 -crystal. L; B/, for some crystal lattice L. (ii) B ˝ B is connected. B/ Â 0 i¤0 Z 0 ˛i . (iv) There is the unique element of weight 0 in B. b/ l, for all b 2 B. b / D i , for all i 2 I . The condition (v) implies that B. / ˝ B has the unique highest weight element of weight , which is u ˝ b by (vi). Further, every element of B.

R=. // D 0 and Nakayama’s lemma implies that K D 0. M / and use (a) again. R=. M / ' M . 46(2). 6 Uniqueness of quasihereditary covers. Now we state the uniqueness result in [40]. 48. Suppose that C and C 0 are highest weight categories over R. Let f. / j 2 ƒg and f. 0 / j 0 2 ƒ0 g be standard objects of C and C 0 , respectively. We say that C and C 0 are equivalent highest weight categories if there is an equivalence C ' C 0 and a bijection ƒ ' ƒ0 such that if corresponds to 0 then . / goes to .

L; A/˝R F ! A˝R F is a monomorphism. 0 Thus if we write 0 ! L; A/ ! A ! L0 ! F; L / D 0 0. 3], and L is R-split as desired. The other implication is clear. 35. F / D A ˝R F - mod. Suppose that a collection of R-free objects f. R/ is given. F / is a split highest weight category whose standard objects are f. / ˝R F j 2 ƒg, and whose projective objects are  ˝R F -filtered, then . / is R-split, for all . As we only apply the result to O, we add the assumption that projective objects are -filtered, but this is in fact automatic.