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 components Nonassociativity Bibliography

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Shivaji [CastroShivaji(1988)] that examples like the one above can only be found in dimension 1! 15) in B cannot have a nontrivial nonnegative solution that vanishes somewhere in B. 13) must be positive. ; Proof. The theorem will be proved by making use of the moving plane method. , a vector u £ RN with \v\ = 1, and each A G R1, we define a{u) = inf x • v\ Tx = {x G RN : x • v = A}; fi^ = {x G fi : x • u < A}. The Moving Plane Method 31 Therefore Tx is a hyperplane in RN perpendicular to v, Qx is a part of fi lying on one side of Tx, and when A = a(i/), then Tx touches Ct at exactly one point on its boundary.

A vector u £ RN with \v\ = 1, and each A G R1, we define a{u) = inf x • v\ Tx = {x G RN : x • v = A}; fi^ = {x G fi : x • u < A}. The Moving Plane Method 31 Therefore Tx is a hyperplane in RN perpendicular to v, Qx is a part of fi lying on one side of Tx, and when A = a(i/), then Tx touches Ct at exactly one point on its boundary. Let R1^ be the reflection map in RN in the hyperplane Tx: R\{x) = x + 2(A - x • v)v, x e RN. We will use the notation x\ = Rl(x), (niY = RWJ. Under the above assumptions, if A — a(v) is positive and small, (fi^)' fl.

Step 1. 1) in [v, w]. Set Qi := {x 6 fl : «i(a;) > ^ ( z ) } - We show that fii must be empty. , — 0 on SfiiDfi. Therefore, by the maximum principle (applied to every component of Qi), we obtain ui < u2 in fii, contradicting its definition. Hence Qi is empty and ui < 112 in Q,. We can similarly prove that 112 < ui- Therefore wi = ^2Step 2. 6) has a unique solution w* in [w,w]. 6) has at most one solution in [tD,w]. 6) is similar. Let e(x) be the unique solution of —Lu = 1 in il, Bu = 0 = 0 on IV Then by the strong maximum principle, we know that e(x) > 0 on Q.

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